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Topics on dynamic limit order book and its related computation

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Content TOPICS ON DYNAMIC LIMIT ORDER BOOK AND ITS RELATED COMPUTATION by Man Luo ADissertationPresentedtothe FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) August 2021 Copyright 2021 Man Luo Dedication To My Family ii Acknowledgments First and foremost, I would like to express my gratitude to my Ph.D. advisor Professor Jin Ma for accepting me as his Ph.D. student. It is great pleasure to work with him in the past 6 years. He taught me how to cope with setbacks and showed me what a talented, dedicated mathematician looks like. Also, his kindness made me feel not lonely in this foreign country. Under his training, I have definitely become a better version of myself IwouldliketoexpressmyspecialgratitudetoProfessorJianfengZhang. Ihave learnt a lot from him about BSDEs and weak formulation problem. And I want to say thanks to Professor Xin Tong, who helped me find my confidence when I lost it. Besides, many thanks go to Professor Remigijus Mikulevicius, Professor Sergey Lototsky for being the members of my qualifying exam. Also, I want to say thanks to Professor Susan Montgomery and Amy Yung for their kind help during my Ph.D. program. In addition, I’m indebted to Professor Qi Feng and Professor Zhaoyu Zhang for our work deep signature FBSDE algorithm. I had a wonderful time during this project. iii Along the path to the Doctor title, there are many people I need to express my deepest gratitude while I never had a chance to do that. I first want to thank my uncle Chao Luo, who couched me since my undergrad. Without him, I could never choose this path and made all these accomplishments. I do appreciate my Uncle for always being there for me. I also want to express my gratitude to my seniors Jia Zhuo, Zhe Zhang, Haiyang Wang, who played an important role when I made the decision to pursue Ph.D. at USC. Also thanks to my seniors Cong Wu, Weisheng Xie, Rentao Sun, Jian Wang, who helped me a lot while I was trying to find a quant job. Here I also want to say thanks to my best friend Yi Zhao, who gave me many support since my high school and I miss our time at USC. I’m also indebted to my classmates Bowen Gang, Jiaowen Yang, Jiajun Luo, Zimu Zhu, Pengbin Feng, Yusheng Wu, Linfeng Li, Ying Tan, Jian Zhou and Lang Wang for their warm-hearted helps and jow we shared together. Along the road to quant finance world, I also greatly appreciate my undergrad senior Danning Huang, who broadened my view towards the mathematical finance world and industry. As for the life in industry, I’m glad that I have met my intern manager Bernhard Hientzsch from WellsFargo, who taught me a lot about deep BSDE method and my friend Xiaodong Chen, who shared lots of useful industry information to me and showed me how an excellent quant looks like. Last but not least, my deepest love goes to my parents Debao Luo, Xianlian Lin and my lovely wife Wenqian Wu. Without my parents’ unconditional support, iv Icouldn’tgothisfarwhilealwayspreservingapositiveattitude. Theydeservemy special thanks for providing me the best education, freedom to grow up happily. Also to my wife Wenqian, I’m excited that we made it and graduated from Ph.D. program at the same time. To be honest, it’s a quite long and hard journey, but I’m lucky to have you there. Without your accompany, I don’t believe I can make it. Let’s keep fighting, Fight on !!! v Table of Contents Dedication ii Acknowledgments iii Abstract viii Chapter 1: The Equilibrium Model of Limit Order Book (LOB) 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Dynamics of LOB and its density . . . . . . . . . . . . . . . 2 1.1.2 Semi-Markov Process . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 Dynamics of incoming orders . . . . . . . . . . . . . . . . . 6 1.2 Optimal Execution Problem . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Properties of the Value Function . . . . . . . . . . . . . . . . . . . . 13 1.3.1 Continuity of the value function on x, k, q . . . . . . . . . . 17 1.3.2 Continuity of the value function on t . . . . . . . . . . . . . 19 1.3.3 Continuity of w . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.4 Dynamic Programming Principle . . . . . . . . . . . . . . . . . . . 33 1.5 HJB Equation and its Viscosity Solution . . . . . . . . . . . . . . . 37 1.5.1 HJB Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.5.2 Viscosity Solution . . . . . . . . . . . . . . . . . . . . . . . . 41 1.5.3 Constrained Viscosity Solution. . . . . . . . . . . . . . . . . 45 Chapter 2: On the Dynamic Frontiers of the Limit Order Books under Equilibrium Model 51 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.3 The Sellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.4 Structure of Dynamical LOB and its Frontier . . . . . . . . . . . . 72 2.4.1 Acceptable limit orders . . . . . . . . . . . . . . . . . . . . . 74 2.4.2 Frontier and the Good Deal Bound . . . . . . . . . . . . . . 78 2.5 The Connection to the Acceptable Index and BSDEs . . . . . . . . 80 2.6 A Principal-Agent Problem View of the Frontier . . . . . . . . . . . 86 vi 2.7 A Solved Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Chapter 3: Deep Signature FBSDE Algorithm 109 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.2 Algorithms and convergence analysis . . . . . . . . . . . . . . . . . 112 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.3.1 Best Ask Price for GBM European Call Option . . . . . . . 117 3.3.2 Lookback Option Example . . . . . . . . . . . . . . . . . . . 119 3.3.3 EuropeanCallOptionintheHestonModelunderParameter Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.3.4 High Dimensional and Non-linear Example . . . . . . . . . . 126 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.5.1 Signature and signature transformation . . . . . . . . . . . . 130 3.5.2 Backgrounds of numerical examples . . . . . . . . . . . . . . 132 3.5.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Bibliography 148 vii Abstract In this dissertation, we will discuss three projects. The first two projects are joint work with my Ph.D. advisor Professor Jin Ma on topics related to Limit Order Book. In the first project, we study the optimal execution problem on an order driven market under an equilibrium model for the limit order book (LOB). It is afollow-upprojectof[41]. Intheiroriginalmodel,theyassumethattheshape of the LOB can be determined endogenously by an expected return function via a competitiveequilibrumargumentanda↵ectedmainlybythemid-price X andtotal liquidity Q, which is driven by a markovian process. However, by stylized facts, a Poisson process may not be a good fit to the real data, thus, we use a semi-markov processtomodel theliquidityoftheorderbook. Wefirstinvestigatetheregularity ofthevaluefunctionandvalidatethedynamicprogrammingprinciple(DPP)based on the idea of the backward markovization technique. As a main result, we show thattheoptimizationproblemundersemi-markovprocessisaconstrainedviscosity solution to the corresponding Hamilton-Jacobi-Bellman equation. viii In the second project, we try to study the dynamic characterization of the frontier, or the bid(ask) price, of a limit order book (LOB). By bining a model for the discounted cash flow model and the so-called No Good Deal Bounds, we prove that the best bid (ask) price can be characterized as a dynamic risk mea- sure. Furthermore, we show that the two determining factors of the frontier: the holders/sellers of the stock and the issuing firm, form a non-zero sum Stackelberg, collaborative stochastic di↵erential game, and the dynamics of the frontier can be determined by the solution of a principal-agent-type of problem, with the help of the tools such as Backward Stochastic Di↵erential Equation (BSDE). The results of this paper could play a fundamental role in determining the shape of the LOB as well as the liquidity cost in an optimal execution problem, along the lines of [52]. The third project is joint work with Professor Qi Feng and Zhaoyu Zhang. We propose a deep signature/log-signature FBSDE algorithm to solve forward- backward stochastic di↵erential equations (FBSDEs) with state and path depen- dent features. By incorporating the deep signature/log-signature transformation into the recurrent neural network (RNN) model, our algorithm can significantly shorten the training time, improve the accuracy, and extends the time horizon comparing to methods in the existing literature. ix Chapter 1 The Equilibrium Model of Limit Order Book (LOB) 1.1 Introduction Alimitorderbookisarecordofunexecutedlimitorders. Conventionally,thereare two types of orders in the market: market orders and limit orders. Market orders are sent by traders that are willing to either buy or sell the asset immediately. They are often executed at the best prices. Limit orders are posted by traders who are willing to wait until the specified prices are satisfied. Limit orders are not always guaranteed to be executed. We are interested in studying the dynamic movements of the LOB in continuous time. These include the dynamics of both the frontier and the shape of the LOB. There have been large number of literature on these two issue, we shall follow the idea of the so-called equilibrium model, initiated by [41], to analyze the overall movement of the LOB as well as its shape. In particular, we shall adopted the theory established in the recent works of [41] and [51] to identify the important 1 relations among the equilibrium price, the frontier, and the density (or the shape) of the LOB. More precisely, in [41] it was shown that, assuming that a competitive equilibrium price exists among all the prices as a Bertrand-type game with the cost of waiting, then the shape of the LOB can be determined endogenously by the equilibrium price, which depends on the fundamental price (or mid-price) of the stock, and the total liquidity of the LOB. 1.1.1 Dynamics of LOB and its density In this subsection we recall the main elements in the dynamic equilibrium model proposed in [41], which will be the basis of our model as well. Let us denote X = {X t } t 0 to be the fundamental value of the asset, which we shall use as the “mid-price”; and Q ={Q t } t 0 to be the total liquidity, or the overall volume of the LOB. We also assume that the equilibrium is “quantified” by a common expected utility on each price, which depends on the fundamental price and the total liquidity, and is denoted by U(X,Q). The main argument in [41] is that, if a (large) market buy order comes in and ↵ shares of the stock were purchased, where ↵ 2 (0,Q], then the lowest portion of ↵ shares of the LOB will be consumed. Thus, if we denote p(0) =p(0,X,Q), to be the lowest ask price, then there exists a unique p(↵ )>p(0) such that: Z p(↵ ) p 0 µ ⇤ (X,Q,y)dy = ↵. 2 On the other hand, we assume that, in equilibrium, the average price of the sold block should have the same expected return of the remaining orders in the LOB, which has a total of Q ↵ shares after the purchase. In other words, we assume that: for any ↵ that 0 ↵  Q,wehave: 1 ↵ Z p(↵ ) p 0 yµ ⇤ (X,Q,y)dy =U(X,Q ↵ ). Solving (1.1) and (1.1) we obtain 8 > > < > > : p(↵ )=U(X,Q ↵ ) ↵ @U @x U(X,Q ↵ ); µ(p(↵ )) = 1 p 0 (↵ ) . We note that by setting ↵ =0,wehave p(0,X,Q)=U(X,Q) That is, the ’frontier’ of the LOB is exactly the representative of the equilibrium, as expected. Furthermore, if we assume that p(↵ ) is invertible and denote h(y)= p 1 (y), then we can have: µ(y)= ⇣ h(y) @ 2 U @Q 2 (X,Q h(y)) 2 @U @Q (X,Q h(y)) ⌘ 1 3 We shall assume that the movement of the LOB depends solely on the investment activites, namely the investor herself, and all other investors(buyers and sales). ’The investor’ is the particular person who will carry out the optimal execution problem later. 1.1.2 Semi-Markov Process The semi-Markov processes were introduced independently and almost simultane- ously in [61], [68] and [69] in 1954-1955 as a generalization of Markov Chain. One can also find a detailed introduction about semi-markov process in [46]. Essen- tially, Semi-Markov processes are a generalization of Markov processes since the exponentialdistributionoftimeintervalsisreplacedwithanarbitrarydistribution. In [61] and [68] semi-markov process was introduced in order to reduce the limita- tion induced by the exponential dsitribution of the corresponding time intervals. In [59] a general definition of stepped semi-Markov processes was proposed and also it provided the first version of the Kolomogorov’s equation for semi-Markov processes in the form of an integro-di↵erential Volterra equation. Let ( S, )be a state space where is a -algebra on S,generatedfromthespace S endowed with the discrete topology induced by the matrix: ⇢ = ⇢ 1,x6=y 0, x = y 4 Then let’s consider on S right-continuous processes X(t), t 0, whose paths are stepped functions. Hence the paths are functionst7! x(t), such that for anyt 0 there exists a> 0suchthatforallh2 (0, )it’struethatx(t)=x(t+h),i.e. the functions t7! x(t)areright-continuousinthediscretetopology,andfurtherthey have a finite number of discontinuities on any finite interval of time. Obviously the paths have a finite number of discontinuities on any finite interval(of time). Processes with these paths are semi-markov processes in the sense of [36] chapter 3, if the couple (X(t), X (t)), where: X (t):=t (0_ sups t :X(s)6=X(t)). is a strict Markov process. Aprocesswiththesepropertiescanbeconstructedasfollows. Let X n be a discrete-time Markov chain on S. And define X(t)=X n , n 1 X i=0 T i  t< n X i=0 T i . where T i = i i 1 and i is the jump time of the i’th jump. Furthermore, we define N(t):=max{n 2 N : P n 1 i=1 T i  t}. Then we can say that it’s equivalent to define X(t):= X N(t) . In the view of [36] [Chapter3, Section3, Lemma2], the processX(t)satisfiesthedefiningpropertiesofsemi-Markovprocess,i.e. thecouple (X(t), X (t)) is a strict(homogeneous) Markov process, where X (t)=t N(t) . 5 At last, we shall point out that a notable special class of semi-Markov process is known as the Renewal Process. We will use this type of semi-markov process to model the investment activities. 1.1.3 Dynamics of incoming orders In the original paper [41], they assume that the activities of other investors are aggregated as a large investor whose investment activities is described by a com- pound process Y t = P Nt i=1 U i ,t 0, where {U i } 1 i=1 is a sequence of i.i.d random variableswithE{|U i |}<1.Theinvestmentactivitiesincludecominglimitorders, market orders and cancellation orders. In my first project, I proposed to use the Renewal process to model the investment activities under the semi-markov set- ting. The main reason that I want to replace the Poisson process is the stylized facts that the arrival of a new book event at the bid or ask and its corresponding inter-arrival time indeed depends on the nature of previous events. So Poisson process is not a good fit. Many researchers then studied other point processes like hawkes process. There are several advantages about the renewal process. The main advantage is that it allows the inter-arrival times between books events(limit orders, market orders, book cancellation) to have an arbitrary distribution. It’s convenient to simulate while it has non parametric estimation. Also it reproduces well microstructure e↵ects, di↵uses on macroscopic scale. And also by backward techniques, we can have a Markov embedding with observable state variables. 6 About the semi-markov setting in our paper, we follow the setting in [4], We use a renewal counting process N = {N t } t 0 to count the investment activities. More precisely, denoting { n } 1 n=1 to be the jump times ( 0 := 0) of N, and T i = i i 1 ,i=1,2,..., to be its waiting times, we assume that T 0 i sareindependent and identically distributed, with a common distribution F :R + 7! R + .Weshall also assume that there exists an intensity function :[0,1)7! [0,1)suchthat ¯ F(t):=P(T 1 >t)=exp{ Z t 0 (u)du}, so that (t)= f(t)/ ¯ F(t),t 0, where f is the density function of T 0 i s.Clearly,if (t)⌘ is a constant, then N becomes a standard Poisson process in [41]. In our paper, we assume that there always exists a constant M such that | (t)| M for any t2 [0,T]. Given a renewal counting process N t ,weshallmodelinvestmentactivitiesas Y t = P Nt i=1 U i ,t 0where {U i } 1 i=1 is a sequence of random variables representing the ”size” of the incoming orders. We assume that {U i } are i.i.d with a common distribution v,andindependentof(N t ,B t ). The process Y t is non-Markovian in general(unless the counting process N t is a Poisson process), however, it can be ”Markovized” by the so-called Backward Markovization technique in [71]. More precisely, we define a new process W t =t Nt ,t 0, 7 that is, the time elapsed since the last jump. Then it’s known that the process (t,Q t ,W t ),t 0 is a piecewise deterministic Markov process(PDMP). We note that at each jump time i ,thejumpsize | W i | = i i 1 = T i and 0 W t  t<T,t2 [0,T]. Throughout this paper we denote{F t } t>0 as the filtration, where F t :=F W t _F B t _F Y t , which will be the basic information allowed in our model. A main subtleties in the study of semi-markov process is the ’delayed’ renewal process. Instead of starting the clock at t=0,westartfrom s 2 [0,T], such thatW s = w,P-a.s. Let us consider the regular conditional probability distribu- tion(RCPD) P sw (·):= P[·|W s = w]on(⌦ ,F), and consider the ’shifted’ version of processes (B,Q,W)onthespace(⌦ ,F,P sw ;F s ), whereF s ={F t } t s .Wefirst defineB s t :=B t B s ,t s.Clearly,B s isanF s -BrownianmotionunderP sw ,define on [s,T], withB s s = 0. Next, we restart the clock at times2 [0,T]bydefiningthe new counting process N s t :=N t N s ,t2 [s,T]. Then underP sw , N s is a ’delayed’ renewal process, in the sense that while its waiting times T s i ,i 2, remain inde- pendent, identically distributed as the original T i ,its’time-to-firstjump’,denoted by T s,w 1 :=T Ns+1 w = Ns+1 s,shouldhavethesurvivalprobability P sw {T s,w 1 >t} =P{T 1 >t+w|T 1 >w} =e R w+t w (u)du . In what follows, we shall denote N s t | Ws=w := N s,w t ,t s, to emphasize the dependence on w as well. Correspondingly, we shall denote Q s,w t = P N s,w t i=1 U i 8 and W s,w t := w +W t W s . It is readily seen that (B s t ,Q s,w t ,W s,w t ),t s, is an F s adapted process defined on (⌦ ,F,P sw ), and it is Markovian. At the end of this section, We should mention that many researchers also used semi-markov process to model the price movement in LOB, like [29] and [75]. It’s also a interesting perspective towards how one can use semi-markov process. 1.2 Optimal Execution Problem The optimal execution problem can stated as: given K share numbers, can we designanoptimalstrategytoselltheseKshares? Weconsiderafinitetimehorizon [0,T]. For simplicity, we assume that there is only one stock traded in an order drivenmarket,andtheinterestrateis0. Wefirstgivethemathematicaldescription of the basic elements involved in our model. 1. Fundamental Price. We assume that the underlying stock has a funda- mental value (or mid-price) which is known to the public. But the market price deviates away from it, due to the possible illiquidity, which leads to the bid-ask spread. Since the fundamental value only a↵ects our model as a source of random- ness, we simply assume that it’s a di↵usion, and satisfies the following stochastic di↵erential equation(SDE): X t =x+ Z t 0 b(s,X s )ds+ Z t 0 (s,X s )dB s ,t 0. 9 where b and satisfy the following standing assumptions (H1):(i)b and are deterministic functions, continuous in t, and uniformly Lipschitz continuous in x, with a common uniform Lipschitz constantL> 0. (ii)x> 0, (t,0) = 0, and b(t,0) 0. In the rest part of the paper, we assume the fundamental price follows a geo- metric brownian motion such that dX t =bX t dt+X t dB t . 2. The Limit Order Book(LOB) We assume that there are patient and impatient investors in the market, and they put di↵erent bid and/or ask prices to either liquidate or purchase the given stock based on their preferences. Since in this paper we consider the optimal execution problem for purchasing the stock, only the sell side LOB will be relevant. We thus assume in what follows that all the buyers are impatient and only make “market orders” (i.e., buying whatever is available on the market), and consequently there is no ”buy side” LOB. Moreover, we isolate one particular investor, referred as the investor, who will carry out the optimal execution problem later. We shall assume that the movement of the LOB depends on solely on the investment activities, as we discussed earlier we use a semi-markov process Y w,q t to model these activities. 10 3. TheInventoryProcess We assume that the investor is trying to purchase acertainnumber,sayK,sharesofthegivenstockwithinagiventimehorizon [0,T], and denote the accumulated number of shares up to time t 2 [0,T]by ⇡ t . Then clearly ⇡ = {⇡ t : t  0} is an increasing process, and we assume that it is F predictable. Furthermore, due to technical reasons, we only consider a specific type of trading strategies, that for every ⇡ ,therealwaysexistsatradingintensity function a t such that d⇡ t =a t dt.WeshallassumethereexistsalargeconstantM such that|a t | M for any t2 [0,T]. We thus define A M ad := {⇡ : ⇡ isF-predictable, non-decreasing, and d⇡ t =a t dt,⇡ T  K,|a t | M for any t2 [0,T]}. We can now describe the dynamics of the total number of shares of the stock in (sell)LOB denoted by Q = {Q t : t 2 [0,T]}.Weshallconsiderinthispaperthe simplest case in which the dynamics of Q can be a↵ected by only two factors: the order made by the investor itself, ⇡ ,andtheordersmadebythethelargeinvestor, Y.WeknowthatY is also influenced by the processW,thisimpliesthatQ is also 11 influenced byW.Then,foragivenstrategyandinitialinventoryq,themovement of Q ⇡,w,q is determined by: Q ⇡,w 0 :=q,andfort> 0, 8 > > > < > > > : Q ⇡,w,q t :=Q ⇡,w,q ⌧ i (⇡ t ⇡ ⌧ i ) t2 (⌧ i ,⌧ i+1 ); Q ⇡,w,q ⌧ i+1 := (Q ⇡,w,q ⌧ i+1 + Y ⌧ i+1 ) + ,t = ⌧ i+1 . (1.1) Inarealworld,theQcan’tbenegative,soit’snaturaltoonlyconsiderthefollowing admissible strategies: given q 0: A w,k,q ad [t,T]:= {⇡ 2A M ad :Q ⇡,w,q t 0,for all t2 [0,T],P-a.s., where Q ⇡,w,q is defined by (1.1) and⇡ t =k}. Throughout the paper, we shall denote R + := (0,1), ¯ R + := [0,1), O t :=R + ⇥ [0,K)⇥ R + ⇥ [0,t), ¯ O t := ¯ R + ⇥ [0,K]⇥ ¯ R + ⇥ [0,t]. 4. Value Function In the rest of the paper, given a trading strategy ⇡ ,andfor given (t,x,k,q,w), we denote X := X t,x , Q := Q ⇡,w,q and A w,k,q ad [t,T]:=A ad [t,T] for notation simplicity. Then we can define the cost function as J(t,x,k,q,w;⇡ ):=E tw n Z T t U(X s ,Q s )d⇡ s +g(X T ,K ⇡ T ) o . 12 Naturally, we care about the minimum cost to finish our job V(t,x,k,q,w):= inf ⇡ 2 A ad [t,T] J(t,x,k,q,w;⇡ ). Here we make following assumptions about U. (H2):The expected utility function U : R + ⇥ ¯ R+ ! R + enjoys the following properties: (i)U is non-decreasing in x, and @Q = @U @Q < 0, @ 2 Q = @ 2 Q @Q 2 > 0. (ii) U is uniformly Lipschitz continuous in (x,q), with Lipschitz constantL> 0. 1.3 Properties of the Value Function In this section we present some results about the regularity of the value function V. In order to make the system markovian, we add a random clock W into the sytem, which changes the nature of the dynamics significantly. Many well-known properties of the value function is not that obvious now. We begin this section by introducing several lemmas, which will be useful in the later proof . Lemma 1.3.1. Let (⌦ ,F,P) be a complete probability space, and ⇣ :⌦ ! D m T be a D m valued process. Let F ⇣ t = {⇣ (s):0  s  t}. Then :[0,T]⇥ ⌦ ! X is {F ⇣ t } t 0 adapted if and only if there exists an ⌘ 2A m T (X) such that (t,!)= ⌘ (t,⇣ .^ t (!)), P a.s.-!,8t2 [0,T]. 13 Lemma 1.3.2. For any ⇡ 2A w,k,q ad [0,t] and ⇡ h 2A w,k,q ad [0,t+h], where ⇡ h s = ⇡ s for s2 [0,t]we can have E{|Q ⇡ h ,0,q t+h Q ⇡, 0,q t |} Ch. Proof. LetC> 0 be a generic constant that is allowed to vary from line to line in the proof. E{|Q ⇡ h ,0,q t+h Q ⇡, 0,q t |} E{ 1 X i=1 |U i |1 {t ⌧ i  t+h} }+ch =E{|U i |}E{N t+h N t }+ch. (1.1) By the definition of intensity function E{N t+h N t } = Z t+h t (u)du Then (1.1) becomes E{|Q ⇡ h ,0,q t+h Q ⇡, 0,q t |} E{|U i |}E{N t+h N t }+ch C Z t+h t | (u)|du+ch Ch. Therefor we proved the lemma. Lemma 1.3.3. For the fundamental price, we derive the inequality E{ sup s2 [t,T] |X t,x 2 s X t,x 1 s |} C(|x 2 x 1 |). 14 Proof. LetC> 0 be a generic constant that is allowed to vary from line to line in the proof. It holds that E |X t,x 2 s X t,x 1 s | 2  CE n |x 2 x 1 | 2 + Z s t b(u,X t,x 2 u ) b(u,X t,x 1 u )du 2 + Z s t (u,X t,x 2 u ) (u,X t,x 1 u dB u ) 2 o  CE n |x 2 x 1 | 2 + Z s t b(u,X t,x 2 u ) b(u,X t,x 1 u ) 2 du + Z s t (u,X t,x 2 u ) (u,X t,x 1 u ) 2 du o  CE n |x 2 x 1 | 2 + Z s t |X t,x 1 u X t,x 2 u | 2 du o . Then by applying Gr¨ onwall’s inequality, it holds that E{ sup s2 [t,T] |X t,x 2 s X t,x 1 s | 2 }  CE{|x 2 x 1 | 2 }+C Z T t E{|X t,x 1 s X t,x 2 s | 2 }ds  C|x 2 x 1 | 2 . Then by H¨ older inequality, it holds that E{ sup s2 [t,T] |X t,x 2 s X t,x 1 s |} E{ sup s2 [t,T] |X t,x 2 s X t,x 1 s | 2 } 1 2  C|x 2 x 1 |. We thus verified the lemma. Lemma 1.3.4. For t 2 >t 1 , it holds that E{|X t 1 ,x t 2 x|} C(1+|x|) p t 2 t 1 . 15 Proof. LetC> 0 be a generic constant that is allowed to vary from line to line in the proof. It holds that E{|X t 1 ,x t 2 x| 2 } = E n Z t 2 t 1 b(t,X t )dt+ (t,X t )dB t 2 o  CE n ( Z t 2 t 1 b(t,X t )dt) 2 +( Z t 2 t 1 (t,X t )dB t ) 2 o  CE n Z t 2 t 1 b(t,X t ) 2 dt+ Z t 2 t 1 (t,X t ) 2 dt o (1.2)  CE n Z t 2 t 1 LX 2 t dt+ Z t 2 t 1 LX 2 t dt o  CE{(1+|x|) 2 (t 2 t 1 )} =C(1+|x|)(t 2 t 1 ). By H¨ older inequality, we derive from (1.2) that E{|X t 1 ,x t 2 x|} C(1+|x|) p (t 2 t 1 ). In the rest of the paper, we will apply lemma 1.3.1 - lemma 1.3.4 many times to prove the regularity of the value function. We will first show the regularity with respect to the fundamental price x,quantity q and the purchased shares k. 16 1.3.1 Continuity of the value function on x, k, q Proposition 1.3.5. For each t2 [0,T] and w2 [0,t]. The value function is non- decreasing in x, non-increasing in k and q, respectively, and uniformly Lipschitz continuous with respect to (x,k,q)2 ¯ O. Proof. LetC> 0 be a generic constant that is allowed to vary from line to line in the proof. We first check the properites in x. Assume x 1 <x 2 .Forfixed (t,k,q,w), we denote X =X x =X t,x , and Q =Q ⇡,w,q for simplicity. Then by the comparison theorem of SDE, we have X x 1 s  X x 2 s ,forall t s T,P-a.s. Since both U and g are non-decreasing and uniformly Lipschitz continuous in x,forany ⇡ 2A w,k,q ad [t,T]weseethat: 0  J(t,x 2 ,k,q,w;⇡ ) J(t,x 1 ,k,q,w;⇡ ) = E tw n Z T t [U(X x 2 s ,Q s ) U(X x 1 s ,Q s )]d⇡ s +g(X x 2 T ,K ⇡ T ) g(X x 1 T ,K ⇡ T ) o  CE tw sup s2 [t,T] |X x 2 s X x 1 s |  C(x 2 x 1 ). (1.3) Byswitchingtheroleofx 1 andx 2 ,wecaneasilydeducetheLipschitzpropertyinx. Next, for any (t,x,q,w), we denoteA k ad [t,T]:=A w,k,q ad [t,T], and Q ⇡ := Q ⇡,w,q for notation simplicity. We check the properties in k.Let0 k 1 <k 2  K. For any ⇡ 2A k 1 ad [t,T], consider the strategy ⇡ 0 s := [k 2 +(⇡ s k 1 )]^ K,s2 [t,T]. Clearly, ⇡ 0 2A k 2 ad [t,T], and it satisfies Q ⇡ 0 Q ⇡ ,J(t,x,k 2 ,q,w;⇡ 0 )  J(t,x,k 1 ,q,w;⇡ ) 17 and thus V(t,x,k 2 ,q,w)  V(t,x,k 1 ,q,w)Ontheotherhand,foranystrategy ⇡ 2A k 2 ad [t,T], let ⇡ 0 := ⇡ (k 2 k 1 )2A k 1 ad [t,T]. Then Q ⇡ 0 =Q ⇡ ,andthus: J(t,x,k 1 ,q,w;⇡ 0 ) J(t,x,k 2 ,q,w;⇡)= E tw g(X T ,K ⇡ 0 T ) g(X T ,K ⇡ T )  C(k 2 k 1 ). Obviously, we can also get uniform Lipschitz continuity of V in k.Itremains to prove the Lipschitz property in q. As before we first assume 0  q 1 0, it holds that lim h!0 |V(t,x,k,q,w) V(t+h,x,k,q,w)|=0. Proof. LetC> 0 be a generic constant that is allowed to vary from line to line in the proof. We note that the clock process W t,w s cannot be controlled, thus it’s not possible to keep the process W frozen at the initial state w during the time interval [t,t+h] by any control strategy. We shall try to get around this by adopting the idea of time shift so as to freeze the w-clock. To be more precise, let us assume that t=0and w=0,othercasescanbe argued similarly. For h 2 (0,T), let ⇡ h 2A w,k,q ad [h,T]. We define ¯ ⇡ h t = ⇡ h t+h ,t 2 [0,T h]. Then ¯ ⇡ h is adapted to the filtration ¯ F h :={F t+h } t 0 .Considertheopti- mization problem on the new probability set-up (⌦ ,F,P h0 , ¯ F h ; ¯ B h , ¯ Q h , ¯ W h ), where ( ¯ B h , ¯ Q h , ¯ W h )=(B h h+t ,Q h h+t ,W h h+t ),t 0. Let us denote the corresponding admis- sible control set by ¯ A 0,k,q ad [0,T h]. Then ¯ ⇡ h 2 ¯ A 0,k,q ad [0,T h], and corresponding fundamental price and investment activities denoted by ¯ X t and ¯ Q t should satisfy the SDE: ¯ X t =x+ Z t 0 µ ¯ X s ds+ Z t 0 ¯ X s d ¯ B s ,t 0. 20 ¯ Y t = ¯ Nt X i=0 U i . 8 > > > < > > > : ¯ Q ¯ ⇡ h ,0,q t := ¯ Q ¯ ⇡ h ,0,q ⌧ i (¯ ⇡ h t ¯ ⇡ h ⌧ i ) t2 (⌧ i ,⌧ i+1 ); ¯ Q ¯ ⇡ h ,0,q ⌧ i+1 := ( ¯ Q ¯ ⇡ h ,0,q ⌧ i+1 + ¯ Y ⌧ i+1 ) + ,t = ⌧ i+1 . (1.4) Since the SDE and jump process is obviously pathwisely unique, whence unique in law, we see that the laws of { ¯ X t } t 0 ,{ ¯ Q t } and that of {X t+h } t 0 and {Q t+h } underP h0 , are identical. In other words, we specify the time duration in the cost functional, then we should have J(h,x,k,q,0,⇡ h ):= E h0 n Z T h U(X h,x s ,Q ⇡ h ,0,q s )d⇡ h s +g(X h,x T ,K ⇡ h T ) o = E h0 n Z T h 0 U( ¯ X s 0,x , ¯ Q s ⇡ h ,0,q )d¯ ⇡ h s +g( ¯ X 0,x T h ,K ¯ ⇡ h T h ) o =: ¯ J 0,T h (0,x,k,q,0,¯ ⇡ h ). therefore, we can have our value function as V(h,x,k,q,0) = inf ¯ ⇡ 2 ¯ A 0,k,q ad [0,T h] ¯ J 0,T h (0,x,k,q,0;¯ ⇡ ). 21 Now, for the given ¯ ⇡ h 2 ¯ A 0,k,q ad [0,T h]wecanfind ⌘ 2A 3 T h (R 2 ), such that ¯ ⇡ h t = ⌘ (t, ¯ B h .^ t , ¯ Q h .^ t , ¯ W h .^ t ),t2 [0,T]. We now define e ⇡ h t := ⌘ (t,B .^ t^ (T h) ,Q .^ t^ (T h) ,W .^ t^ (T h) ),t2 [0,T h]. Then e ⇡ h 2 A 0,k,q ad [0,T]. Furthermore, since the law of ( ¯ B h t , ¯ Q h t , ¯ W h t ),t 2 [0,T h], under P h0 ,andthatof(B t ,Q t ,W t ),t 2 [0,T h], under P,are identical, by the pathwise uniqueness(whence uniqueness in law) of the solu- tions to the corresponding SDE, the process {(X t ,Q e ⇡ h t ,0,q t ,W t ,e ⇡ h t )} t2 [0,T h] and {( ¯ X t , ¯ Q ¯ ⇡ t h ,0,q t , ¯ W t ,¯ ⇡ h t )} t2 [0,T h] are identical in law. For any ⇡ h 2 A 0,k,q ad ([h,T]), we define ⇡ t = ⇡ h t+h for t2 [0,T h], it’s clear that ⇡ 2A 0,k,q ad ([0,T]). Then, J(h,x,k,q,0,⇡ h )= ¯ J 0,T h (0,x,k,q,0;¯ ⇡ h ) = E h0 n Z T h 0 U( ¯ X 0,x t , ¯ Q ¯ ⇡, 0,q t )d¯ ⇡ t +g( ¯ X 0,x T h ,K ¯ ⇡ T h ) o = E 00 n Z T h 0 U(X h,x t+h ,Q e ⇡ t,0,q t+h )de ⇡ t +g(X h,x T ,K e ⇡ T h ) o = E 00 n Z T h 0 U(X h,x t+h ,Q ⇡ h t+h ,0,q t+h )d⇡ h t+h +g(X h,x T ,K ⇡ h T ) o . 22 Then it holds that |J(h,x,k,q,0,⇡ h ) J(0,x,k,q,0,⇡ )| = E 00 n Z T h 0 [U(X h,x s+h ,Q s+h ⇡ h s+h ,0,q ) U(X 0,x s ,Q ⇡ h s+h ,0,q s )]d⇡ h s+h +[g(X h,x T ,K ⇡ h T ) g(X 0,x T ,K ⇡ h T )] o  CE 00 { sup s2 [0,T h] |X h,x s+h X 0,x s |+sup s2 [0,T h] |Q ⇡ h ,0,q s+h Q ⇡ h ,0,q s |} +CE 00 {|X h,x T X 0,x T |}. ForE 00 n sup s2 [0,T h] X h,x s+h X 0,x s o ,wecanrewriteX h,x s+h X 0,x s =X h,x s+h X 0,x s+h + X 0,x s+h X 0,x s .Thenwecanderivethat E 00 n sup s2 [0,T h] X h,x s+h X 0,x s o  E 00 n sup s2 [0,T h] X h,x s+h X 0,x s+h o +E 00 n sup s2 [0,T h] X 0,x s+h X 0,x s o . By Lemma 1.3.3 for any s2 [0,T h], it holds that E 00 {|X h,x s+h X hX 0,x h s+h |} CE 00 {|X 0,x h x|} C(1+|x|) p h. (1.5) Also from the proof in Lemma 1.3.4, it also holds that E 00 n sup s2 [0,T h] X 0,x s+h X 0,x s o  E 00 n sup s2 [0,T h] Z s+h s b(u,X s,X 0,x s u )du+ Z s+h s (u,X s,X 0,x s u )dB u o  C(1+|x|) p h. (1.6) 23 Combining equation 1.5 and equation 1.6, it holds that E 00 n sup s2 [0,T h] X h,x s+h X 0,x s o  C(1+x) p h. (1.7) ForE 00 {|X h,x T X 0,x T |},withsameargument,wecanhave E 00 {|X h,x T X 0,x T |} C(1+x) p h. (1.8) Lastly, we need to estimateE 00 {|Q ⇡ h ,0,q s+h Q ⇡ h ,0,q s |},byLemma1.3.2,itholdsthat E 00 {|Q ⇡ h ,0,q s+h Q ⇡ h ,0,q s |} Ch. (1.9) Now combining equation 1.7 - 1.9, we showed that for any ⇡ h 2 A w,k,q ad [t + h,T], there exists a corresponding strategy ⇡ 2 A w,k,q ad [t,T]such that lim h!0 |J(t,x,k,q,w;⇡ ) J(t + h,x,k,q,w;⇡ )| =0. Conversely,forany ⇡ 2 A 0,k,q ad [0,T], we can write the value function as J(0,x,k,q,0;⇡)= E 00 n Z T h 0 U(X 0,x s ,Q ⇡, 0,q s )d⇡ s + Z T T h U(X T h,X 0,x T h s ,Q ⇡,W T h ,Q T h s )d⇡ s . +g(X 0,x T ,K ⇡ T ) o . 24 Now define ⇡ h t = ⇡ t h for any t2 [h,T]. Then we denote I1:=E 00 n Z T h 0 U(X 0,x s ,Q ⇡, 0,q s )d⇡ s +g(X 0,x T ,K ⇡ T ) o . and I2:=J(h,x,k,q,0;⇡ h )=E h0 n Z T h U(X h,x s ,Q ⇡ h ,h,q s )d⇡ h s +g(X h,x T ,K ⇡ h T ) o . I3:=E 00 n Z T T h U(X T h,X 0,x T h s ,Q ⇡,W T h ,Q T h s )d⇡ s o . With similar argument in the earlier proof in this theorem, we can conclude that |I1 I2| C(1+|x|) p h. (1.10) Lastly, we need to estimate I 3 .Since d⇡ t =a t dt,then I 3 = E 00 n Z T T h U(X T h,X 0,x T h s ,Q ⇡,W T h ,Q T h s )a t dt o  Ch. (1.11) Combine equation 1.10 and 1.11, we derived the desired result. 25 1.3.3 Continuity of w Next we will present the regularity with respect to the random clock w.Tobegin with, we need to introduce to intermediate cost functions J 1 and J 2 first. Then we can decompose the cost function J as J 1 and J 2 . J 1 (t,x,k,q,w;⇡ ):=E tw n Z T t U(X t,x s ,Q ⇡,w,q s )d⇡ s +g(X t,x T ,K ⇡ T )|T t,w 1 >h o . J 2 (t,x,k,q,w;⇡ ):=E tw n Z T t U(X t,x s ,Q ⇡,w,q s )d⇡ s +g(X t,x T ,K ⇡ T )|T t,w 1 <h o . V 1 (t,x,k,q,w):= inf ⇡ 2 A w,k,q ad [t,T] J 1 (t,x,k,q,w;⇡ ). We first show that value function V(t+h,x,k,q,w+h)canbeapproximatedby V 1 (t,x,k,q,w). Then with the help of continuity of temporal variable t,wecan prove V(t,x,k,q,w) is also continuous with respect to random clock w. Lemma 1.3.7. For anyh> 0, it holds that lim h!0 V 1 (t,x,k,q,w) V(t + h,x,k,q,w+h) =0. 26 Proof. LetC> 0 be a generic constant that is allowed to vary from line to line in the proof. For any ⇡ 2A w,k,q ad [t,T], denote q 1 := R t+h t d⇡ s ,itholdsthat J 1 (t,x,k,q,w;⇡):= E tw n Z T t U(X t,x s ,Q ⇡,w,q s )d⇡ s +g(X t,x T ,K ⇡ T )|T t,w 1 >h o = E tw n Z t+h t U(X t,x s ,Q ⇡,w,q s )d⇡ s + Z T t+h U(X t+h,X t,x t+h s ,Q ⇡,W t+h ,Q t+h s )d⇡ s +g(X t,x T ,K ⇡ T )|T t,w 1 >h o = E tw n Z t+h t U(X t,x s ,Q ⇡,w,q s )d⇡ s |T t,w 1 >h o +E tw n J(t+h,X t,x t+h ,k+q 1 ,Q ⇡,w +h,q q 1 t+h ,w+h;⇡ ) o . We denote I 1 and I 2 as I 1 :=E tw n Z t+h t U(X t,x s ,Q ⇡,w,q s )d⇡ s |T t,w 1 >h o , I 2 :=E tw n J(t+h,X t,x t+h ,k+q 1 ,Q ⇡,w +h,q q 1 t+h ,w+h;⇡ ) o . Now we define ⇡ h s = ⇡ s q 1 for any s2 [t+h,T]. It’s clear that ⇡ h 2A w+h,k,q ad [t+ h,T]. Then we can get |I 2 J(t+h,x,k,q,w+h;⇡ h )| = E tw n J(t+h,X t,x t+h ,k+q 1 ,Q ⇡,w +h,q q 1 t+h ,w+h;⇡ ) J(t+h,x,k,q,w+h;⇡ h ) o  CE tw {|X t,x t+h x|+q 1 }  C(1+|x|) p h+Ch. (1.12) 27 Therefore, when h is small enough, we have |I 2 J(t+h,x,k,q,w+h;⇡ h )|=0. Moreover, we can write I 1 as |I 1 | = E tw n Z t+h t U(X t,x s ,Q ⇡,w,q s )a s ds|T t,w 1 >h o  Ch. (1.13) Combining (1.12) and (1.13), we can conclude that for any ⇡ 2A w,k,q ad [t,T], we can always find a corresponding ⇡ h 2A w,k,q ad [t+h,T]suchthat lim h!0 |J 1 (t,x,k,q,w;⇡ ) J(t+h,x,k,q,w+h;⇡ h )|=0. (1.14) Now consider any ⇡ h 2A w+h,k,q ad [t+h,T]. We define ⇡ s = ⇡ h s during time interval [t+h,T]and ⇡ s ⌘ k during time interval [t,t+h]. It’s clear that ⇡ 2A w,k,q ad [t,T]. Then we can have J 1 (t,x,k,q,w;⇡):= E tw n Z T t U(X t,x s ,Q ⇡,w,q s )d⇡ s +g(X t,x T ,K ⇡ T )|T t,w 1 >h o = E tw n Z T t+h U(X t+h,X t,x t+h s ,Q k,W t+h ,q s )d⇡ s +g(X t,x T ,K ⇡ T )|T t,w 1 >h o = E tw n J(t+h,X t,x t+h ,k,Q k,w+h,q t+h ,w+h;⇡ h ) o . With similar arguments, we can conclude that for any ⇡ h 2A w+h,k,q ad [t+h,T], we can always find a ⇡ 2A w,k,q ad [t,T]suchthat lim h!0 |J 1 (t,x,k,q,w;⇡ ) J(t+h,x,k,q,w+h;⇡ h )|=0. (1.15) 28 By (1.14) and (1.15), we proved the lemma. Before we come to the final step about the continuity of w,westillneedone more intermediate step. We show the continuity of V on the variable (t,w). Proposition 1.3.8. For anyh> 0, we can derive lim h!0 |V(t,x,k,q,w) V(t+h,x,k,q,w+h)|=0. Proof. LetC> 0 be a generic constant that is allowed to vary from line to line in the proof. By the definition of V(t,x,k,q,w), for any✏> 0, there always exists a ⇡ ,wherewedenote q 1 = R t+h t d⇡ s and satisfies ✏+V(t,x,k,q,w) J(t,x,k,q,w;⇡ ) E tw n Z T t U(X t,x s ,Q ⇡,w,q u )d⇡ s +g(X t,x T ,K ⇡ T )|T t,w 1 >h o P tw (T t,w 1 >h) exp R w+h w (u)du E tw n Z T t+h U(X t+h,X t,x t+h s ,Q k+q 1 ,w+h,q q 1 s )d⇡ s +g(X t,x T ,K ⇡ T )|T s,w 1 >h o . 29 Now for any (t+h,X t,x t+h ,k +q 1 ,Q k+q 1 ,w+h,q q 1 t+h ,w +h)indomainbyProposition 1.3.5, we have ✏+V(t,x,k,q,w) exp R w+h w (u)du J(t+h,X t,x t+h ,k+q 1 ,Q k+q 1 ,w+h,q q 1 t+h ,w+h;⇡ ) exp R w+h w (u)du V(t+h,x,k,q,w+h) Cexp R w+h w (u)du (1+|x|) p h. (1.16) Rearranging (1.16), it holds that ✏+V(t,x,k,q,w)+exp R w+h w (u)du C(1+|x|) p h exp R w+h w (u)du V(t+h,x,k,q,w+h). (1.17) From (1.17), it holds that V(t+h,x,k,q,w+h) V(t,x,k,q,w)  ✏+(1 exp R w+h w (u)du )V(t+h,x,k,q,w+h) +exp R w+h w (u)du C(1+|x|) p h. By sending ✏ ! 0and h ! 0, we can claim that V(t + h,x,k,q,w + h) V(t,x,k,q,w) 0. 30 We observe that for any ⇡ 2A w,k,q ad [t,T], it holds that J(t,x,k,q,w;⇡ ) =E tw n Z T t U(X t,x s ,Q ⇡,w,q s )d⇡ s +g(X t,x T ,K ⇡ T )|T t,w 1 >h) o P tw (T t,w 1 >h) +E tw n Z T t U(X t,x s ,Q ⇡,w,q s )d⇡ s +g(X t,x T ,K ⇡ T )|T t,w 1 <h) o P tw (T t,w 1 <h). Next, we denote V 2 (t,x,k,q,w)as V 2 (t,x,k,q,w):= inf ⇡ 2 A w,k,q ad [t,T] E tw n Z T t U(X t,x s ,Q ⇡,w,q s )d⇡ s +g(X t,x T ,K ⇡ T )|T t,w 1 <h) o . Then we can have V(t,x,k,q,w) V(t+h,x,k,q,w+h) = V 1 (t,x,k,q,w)P tw (T t,w 1 >h)+V 2 (t,x,k,q,w)P tw (T t,w 1 <h) V(t+h,x,k,q,w+h)  V 1 (t,x,k,q,w) V(t+h,x,k,q,w+h) +V 2 (t,x,k,q,w)P tw (T t,w 1 <h). 31 Denote I 1 and I 2 as I 1 :=V 1 (t,x,k,q,w) V(t+h,x,k,q,w+h) I 2 :=V 2 (t,x,k,q,w)P tw (T t,w 1 <h). Now we observe I 2 that |I 2 | =|V 2 (t,x,k,q,w)(1 exp R w+h w (u)du )|. and when h is small enough, lim h!0 I 2 =0. For I 1 ,byLemma1.3.7,italsoholds that lim h!0 I 1 =0. Therefore,wecanconcludethat lim h!0 |V(t,x,k,q,w) V(t+h,x,k,q,w+h)|=0. Combining the propositions in this section, we have the following theorem. Theorem 1.3.9. For anyh> 0, the value function satisifes lim h!0 |V(t,x,k,q,w) V(t,x,k,q,w+h)|=0. 32 1.4 Dynamic Programming Principle In this section, we shall verify our value function is valid for Bellman Dynamic Programming Principle(DPP). We will first show the Bellman principle is valid under a fixed time T,thenextendittothestoppingtimecase. Proposition 1.4.1. For any 0 t 1 <t 2  T and (x,k,q,w)2 ¯ O t 1 , we can have: V(t 1 ,x,k,q,w)= inf ⇡ 2 A w,k,q ad [t 1 ,T] E t 1 w n Z t 2 t 1 U(X t 1 ,x s ,Q ⇡,w,q s )d⇡ s +V(t 2 ,X t 1 ,x t 2 ,⇡ t 2 ,Q ⇡,w,q t 2 ,W t 1 ,w t 2 ) o . (1.1) Proof. LetC> 0 be a generic constant that is allowed to vary from line to line in the proof. Let ¯ V(t 1 ,x,k,q,w)denotetherightsideof(1.1). Wefirstshowthat V(t 1 ,x,k,q,w) ¯ V(t 1 ,x,k,q,w). For any ⇡ 2A w,k,q ad [t 1 ,T], then we can have J(t 1 ,x,k,q,w;⇡ ) = E t 1 w n Z T t 1 U(X t 1 ,x s ,Q ⇡,w,q s )d⇡ s +g(X t 1 ,x T ,K ⇡ T ) o = E t 1 w n Z t 2 t 1 U(X t 1 ,x s ,Q ⇡,w,q s )d⇡ s +E[ Z T t 2 U(X t 2 ,X t 1 ,x t 2 s ,Q ⇡,W t 2 Qt 2 s )d⇡ s +g(X t 2 ,X t 1 ,x t 2 T ,K ⇡ T )|F t 2 ] o = E t 1 w n Z t 2 t 1 U(X t 1 ,x s ,Q ⇡,w,q s )d⇡ s +J(t 2 ,X t 1 ,x t 2 ,⇡ t 2 ,Q ⇡,w,q t 2 ,W t 1 ,w t 2 ;⇡ ) o inf ⇡ 2 A w,k,q ad [t 1 ,T] E t 1 w n Z t 2 t 1 U(X t 1 ,x s ,Q ⇡,w,q s )d⇡ s +V(t 2 ,X t 1 ,x t 2 ,⇡ t 2 ,Q ⇡,w,q t 2 ,W t 1 ,w t 2 ) o . This implies that V(t 1 ,x,k,q,w) ¯ V(t 1 ,x,k,q,w). 33 Now it suces to prove the opposite inequality. For any fixed "> 0, we consider a countable partition for the domain of (x,k,q,w)suchthat |x x i | ",k i "  k  k i and q i  q  q i " and w i  w  w i + ✏. Now for each i, we choose ⇡ i such that J(t 2 ,x i ,k i ,q i ,w i ;⇡ i )  V(t 2 ,x i ,k i ,q i ,w i )+ ". Note that ⇡ i k i + k2A w i ,k i ,q i ad [t 2 ,T]⇢A w,k,q ad [t 2 ,T], then by the continuity of the cost function and value function, we can obtain that there exists a constant C such that J(t 2 ,x,k,q,w;⇡ i k i +k)  J(t 2 ,x i ,k i ,q,w;⇡ i )+C"  J(t 2 ,x i ,k i ,q i ,w;⇡ i )+C"  J(t 2 ,x i ,k i ,q i ,w i ;⇡ i )+C"  V(t 2 ,x i ,k i ,q i ,w i )+C"  V(t 2 ,x,k,q,w)+C". Now for any ⇡ 2A w,k,q ad [t 1 ,T], we can define a new strategy ¯ ⇡ as ¯ ⇡ s := ⇡ s 1 [t 1 ,t 2 ] (s)+ ⇥ X i [⇡ i s k i +⇡ t 2 ]1 O i (X t 1 ,x t 2 ,⇡ t 2 ,Q ⇡,w,q t 2 ,W t 1 ,w t 2 ) ⇤ 1 (t 2 ,T] (s). 34 Then we can have V(t 1 ,x,k,q,w)  J(t 1 ,x,k,q,w;¯ ⇡ ) = E t 1 w n Z t 2 t 1 U(X t 1 ,x s ,Q ⇡,w,q s )d⇡ s +E[ Z T t 2 U(X t 1 ,x s ,Q ¯ ⇡,W t 2 ,Qt 2 s )d¯ ⇡ s +g(X t 1 ,x T ,K ¯ ⇡ T )|F t 2 ] o = E t 1 w n Z t 2 t 1 U(X t 1 ,x s ,Q ⇡,w,q s )d⇡ s +J(t 2 ,X t 1 ,x t 2 ,⇡ t 2 ,Q ⇡,w,q t 2 ,W t 2 ;¯ ⇡ t 2 ) o = E t 1 w n Z t 2 t 1 U(X t 1 ,x s ,Q ⇡,w,q s )d⇡ s + X i J(t 2 ,X t 1 ,x t 2 ,⇡ t 2 ,Q ⇡,w,q t 2 ,W t 1 ,w t 2 ;⇡ i k i +⇡ t 2 )1 O i (X t 1 ,x t 2 ,⇡ t 2 ,Q ⇡,w,q t 2 ,W t 1 ,w t 2 ) o  E t 1 w n Z t 2 t 1 U(X t 1 ,x s ,Q ⇡,w,q s )d⇡ s +V(t 2 ,X t 1 ,x t 2 ,⇡ t 2 ,Q ⇡,w,q t 2 ,W t 1 ,w t 2 ) o +C". Since"isarbitrary,wecanconcludethatV(t 1 ,x,k,q,w) ¯ V(t 1 ,x,k,q,w). There- fore, we proved the proposition. To conclude a general version of the dynamic programming principle. Denote T t to be all theF-stopping times taking values in (t,T]. Theorem 1.4.2. For any (t,x,k,q,w)2 [0,T]⇥ ¯ O t and any ⌧ 2T t , it holds that V(t,x,k,q,w)= inf ⇡ 2 A w,k,q ad [t,T] E tw n Z ⌧ t U(X t,x s ,Q ⇡,w,q s )d⇡ s +V(⌧,X t,x ⌧ ,⇡ ⌧ ,Q⌧ ⇡,w,q ,W t,w ⌧ ) o . Proof. For each ⇡ 2A w,k,q ad [t,T]and ⌧ 2T t ,denote I(⇡,⌧ )betheexpectation on the right side. Following the argument in proposition 1.4.1, one can easily show 35 that V(t,x,k,q,w) inf ⇡ 2 A w,k,q ad [t,T] I(⇡,⌧ ). So it suces to show the reversed inequality. We first assume that ⌧ 2T t takes only finitely many valuest<t 1 <···<t m  T. We prove the reversed inequality by induction on m. When m = 1, it follows the proposition 1.4.1 directly. Now assume that it holds for m 1, and that ⌧ takes m values. For any ⇡ 2A w,k,q ad [t,T], we have I(⇡,⌧ )= E tw n Z t 1 t U(X t,x s ,Q ⇡,w,q s )d⇡ +V(t 1 ,X t,x t 1 ,⇡ t 1 ,Q ⇡,w,q t 1 ,W t,w t 1 )1 {⌧ =t 1 } + h Z ⌧ t 1 U(X t,x s ,Q ⇡,w,q s )d⇡ +V(⌧,X t,x ⌧ ,⇡ ⌧ ,Q ⇡,w,q ⌧ ,W t,w ⌧ ) i 1 {⌧>t 1 } o . Notethat{⌧>t 1 }2F t 1 and⌧ takesonlym 1valueson{⌧>t 1 }.Byinductional hypothesis we have I(⇡,⌧ ) = E tw n Z t 1 t U(X t,x s ,Q ⇡,w,q s )d⇡ s +V(t 1 ,X t,x t 1 ,⇡ t 1 ,Q ⇡,w,q t 1 ,W t,w t 1 )1 {⌧ =t 1 } +E ⇥ Z ⌧ t 1 U(X t 1 ,X t,x t 1 s ,Q ⇡,W t,w t 1 ,Q ⇡,w,q t 1 s )d⇡ s +V(⌧,X t,x ⌧ ,⇡ ⌧ ,Q ⇡,w,q ⌧ ,W t,w ⌧ )|F t 1 ⇤ 1 {⌧>⌧ 1 } o E tw n Z t 1 t U(X t,x s ,Q ⇡,w,q s )d⇡ s +V(t 1 ,X t,x t 1 ,⇡ t 1 ,Q ⇡,w,q t 1 ,W t,w t 1 )1 {⌧ =t 1 } +V(t 1 ,X t,x t 1 ,⇡ t 1 ,Q ⇡,w,q t 1 ,W t,w t 1 )1 {⌧>⌧ 1 } o = E tw n Z t 1 t U(X t,x s ,Q ⇡,w,q s )d⇡ s +V(t 1 ,X t,x t 1 ,⇡ t 1 ,Q ⇡,w,q t 1 ,W t,w t 1 ) o V(t,x,k,q,w). 36 where the last inequality comes from the earlier proposition. Since ⇡ is arbitrary we proved the induction. To prove the reversed inequality for arbitrary ⌧ 2T t ,wefirstfindasequence of {⌧ n } where for each ⌧ n 2F t such that ⌧ n ⌧  1 n and ⌧ n #⌧ ,as n!1.By previous arguments we see that the reversed inequality holds for each ⌧ n .That is, V(t,x,k,q,w)  I(⇡,⌧ n )foreach ⇡ 2A w,k,q ad [t,T]. Moreover, by definition of I(⇡,⌧ )wehave I(⇡,⌧ n ) I(⇡,⌧ )= E tw n Z ⌧ n ⌧ U(X t,x s ,Q ⇡,w,q s )d⇡ s +V(⌧ n ,X t,x ⌧ n ,⇡ ⌧ n ,Q ⇡,w,q ⌧ n ,W t,w ⌧ n ) V(⌧,X t,x ⌧ ,⇡ ⌧ ,Q ⇡,w,q ⌧ ,W t,w ⌧ ) o . By the continuity properties of V and ⇡ ,wecanseethattherightsideconverges to 0 as n!1.Thus,weobtainedthat V(t,x,k,q,w)  I(⇡,⌧ )foreach ⇡ 2 A w,k,q ad [t,T]. This implies the result. 1.5 HJB Equation and its Viscosity Solution We are now ready to investigate the main subject of this paper: the Hamilton- Jacobi-Bellman(HJB)equationassociatedtoouroptimizationproblem. Weshould point out that only after the random clock W is brought into the system, there is apossibilitythatwecanhaveaPDEcharacterizationofthevaluefunction. 37 1.5.1 HJB Equation Inthissection,weshallprovethatthevaluefunction,whilenotnecessarilysmooth, is a viscosity solution of the Hamilton-Jacobi-Bellman equation of the optimal execution problem. For any ' 2C 1,2,1,1,1 b ([0,T]⇥ ¯ O t ) denote the set of continuous functions ' on [0,T]⇥ ¯ O t such that the partial derivatives @ t ',@ x ',@ k ',@ q ',@ w ' and @ xx ' existandarecontinuousandbounded. Also, wedefine '(t,x,k,q,w)=0 for (t,x,k,q,w) / 2 [0,T]⇥ ¯ O t .Fornotationsimplicity,wedenote C([0,T]⇥ ¯ O t ):= C 1,2,1,1,1 b ([0,T]⇥ ¯ O t )intherestofthesection. Weintroducethefollowingintegro- di↵erential operators: U[']= Z R ['(t,x,k,(q+u) + ,0) '(t,x,k,q,w)]v(du). and H['](t,x,k,q,w) := H(t,x,k,q,w,',r ',' xx ,U['],⇡ ) := U(x,q)a+a(@ k ' @ q ')(t,x,k,q,w)+(@ t +@ w ' +b@ x ' + 1 2 2 @ xx ')(t,x,k,q,w) + (w)U[']. 38 and L['](t,x,k,q,w) := U(x,q)a+a(@ k ' @ q ')(t,x,k,q,w)+(@ w ' +b@ x ' + 1 2 2 @ xx ')(t,x,k,q,w) + (w)U[']. Moreover, for any (t,x,k,q,w) 2 [0,T]⇥ ¯ O t and ⇡ 2A w,k,q ad [t,T], we denote X := X t,x ,Q := Q ⇡,w,q ,W := W t,w for notation simplicity in the rest of this section.. And for anyF t stopping time ⌧ ,wecandefine I(',⇡,⌧ ):=E tw n Z ⌧ t U(X s ,Q s )d⇡ s +'(⌧,X ⌧ ,⇡ ⌧ ,Q ⌧ ,W ⌧ ) o '(t,x,k,q,w). Lemma 1.5.1. Assume ' 2 C([0,T]⇥ ¯ O t ) and ⌧ is an F t stopping time. Then it holds that I(',⇡,⌧ ^ (t+T t,w 1 )) =E tw n Z ⌧ t H['](s,X s ,⇡ s ,Q s ,W s )ds o . (1.1) Proof. For any F t stopping time ⌧ we denote ˆ ⌧ := ⌧ ^ (t +T t,w 1 ). Let ⇡ 2 A w,k,q ad [t,ˆ ⌧ ], and observe that Q s = q ⇡ s +k, s2 [t,ˆ ⌧ ). Then, it is readily seen that Q t+T s,w 1 =(Q t+T s,w 1 + Y t+T s,w 1 ) + ,andthus 39 '(ˆ ⌧,X ˆ ⌧ ,⇡ ˆ ⌧ ,Q ˆ ⌧ ,W ˆ ⌧ ) '(t,x,k,q,w) = '(ˆ ⌧ ,X ˆ ⌧ ,⇡ ˆ ⌧ ,Q ˆ ⌧ ,W ˆ ⌧ ) '(t,x,k,q,w) +['(t+T s,w 1 ,X t+T s,w 1 ,⇡ t+T s,w 1 ,Q t+T s,w 1 ,0) '(ˆ ⌧ ,X ˆ ⌧ ,⇡ ˆ ⌧ ,Q ˆ ⌧ ,W ˆ ⌧ )]1 {t+T t,w 1  ⌧ } . Then we can denote '(ˆ ⌧,X ˆ ⌧ ,⇡ ˆ ⌧ ,Q ˆ ⌧ ,W ˆ ⌧ ) '(t,x,k,q,w)=I 1 +I 2 1 {t+T t,w 1  ⌧ } . We observe that for I 1 ,thereisnojump,soapplyingItˆ o’s formula we have E tw n '(ˆ ⌧ ,X ˆ ⌧ ,⇡ ˆ ⌧ ,Q ˆ ⌧ ,W ˆ ⌧ ) '(t,x,k,q,w) o = E tw n Z ˆ ⌧ t ⇥ @ t ' +@ w ' +b@ x ' + 1 2 2 @ xx ' ⇤ (s,X s ,⇡ s ,Q s ,W s )ds + Z ˆ ⌧ t ⇥ @ k ' @ q ' ⇤ (s,X s ,⇡ s ,Q s ,W s )a s ds o . For I 2 ,wehave E tw n ['(t+T s,w 1 ,X t+T s,w 1 ,⇡ t+T s,w 1 ,Q t+T s,w 1 ,0) '(ˆ ⌧ ,X ˆ ⌧ ,⇡ ˆ ⌧ ,Q ˆ ⌧ ,W ˆ ⌧ )]1 {t+T t,w 1  ⌧ } o = E tw n Z ⌧ t 0 dF T t,w 1 (s) Z R ['(s,X s ,⇡ s ,(Q s +u) + ,0) '(s,X s ,⇡ s ,Q s ,W s )]v(du) o Recall that dF T t,w 1 (s)= (w) ¯ F T t,w 1 (s)ds = (w)exp R w+s w (u)du ds,and ¯ F T t,w 1 (0) = 1, then we obtain the result. 40 Now by applying dynamic programming principle to value functionV,wehave 0= inf ⇡ 2 A w,k,q ad [t,⌧ ] I(V,⇡,⌧ ). Thenwecanobtainthatinf ⇡ 2 A w,k,q ad [t,⌧ ] E tw n R ⌧ t H[V](s,X s ,⇡ s ,Q s ,W s )ds o =0.By sending ⌧ #t, we can obtain the following HJB equation: {V t +inf a2 [0,M] L[V]}(t,x,k,q,w)=0. (1.2) with the terminal-boundary conditions: V(T,x,k,q,w)=g(x,K k); V(t,x,K,q,w)=0. (1.3) 1.5.2 Viscosity Solution Ingeneral,however,Vmaynotbesmooth. Wethusneedtomakeuseofthenotion of the viscosity solution.Tothisend,letusdenote,for(t,x,k,q,w)2 [0,T)⇥O t , A(t,x,k,q,w):= n ' 2C([0,T]⇥ ¯ O t ):[V '](t,x,k,q,w)=0 o ; A(t,x,k,q,w):= n ' 2A(t,x,k,q,w):V ' attains global max. at (t,x,k,q,w) o ; A(t,x,k,q,w):= n ' 2A(t,x,k,q,w):V ' attains global min. at (t,x,k,q,w) o . 41 Definition 1.5.2. A continuous function V :[0,T]⇥ ¯ O t ! R + is called a viscosity subsolution(resp. supersolution) to the corresponding HJB (1.2) and (1.3) if 1. V(T,x,k,q,w) (resp.  )g(x,K k), andV(t,x,K,q,w) 0 (resp.  0); 2. for any (t,x,k,q,w) 2 [0,T)⇥O t and ' 2 A(t,x,k,q,w) (resp. A(t,x,k,q,w)) one has {' t +inf a2 [0,M] L[']}(t,x,k,q,w) 0(resp. 0). Moreover, V is called a viscosity solution if it’s both a viscosity subsolution and supersolution. Theorem 1.5.3. The value function V of the optimal execution problem is a vis- cosity solution of the corresponding HJB (1.2) and (1.3). Proof. At time T,we cannot purchase anything, so it’s natural that V(T,x,k,q,w)= g(x,K k). If at time t,wehavealreadyhad K shares, then there is no need to purchase anything, it’s obvious that V(t,x,K,q,w)=0. We first prove that viscosity supersolution property. Suppose in the contrary then there exists (t,x,k,q,w)and ' such that c := inf a2 [0,M] {' t +L[']}(t,x,k,q,w)> 0. By applying the DPP, for any> 0, we can find ⇡ such that V(t,x,k,q,w) E tw n Z t+T t,w 1 t U(X s ,Q s )d⇡ s +V(t+T t,w 1 ,X t+T t,w 1 ,⇡ t+T t,w 1 ,Q t+T t,w 1 ,0) o 2 . 42 Now for any> 0, we define the stopping time ⌧ := (t+ )^ (t+T t,w 1 ). Then ⌧ is anF t stopping time. Similar with Theorem 1.4.2, we can show that V(t,x,k,q,w) E tw n Z ⌧ t U(X s ,Q s )d⇡ s +V(ˆ ⌧ 0 ,X ˆ ⌧ 0 ,⇡ ˆ ⌧ 0 ,Q ˆ ⌧ 0 ) o 2 . Then by applying Ito’s formula like we used in Lemma 1.5.1 and property of A(t,x,k,q,w), we can obtain: 2 E tw n Z ⌧ t (' s +L['])(s,X s ,⇡ s ,Q s ,W s )ds o . Since ' is smooth, we deduce that, for is small enough, (' s +L['])(s,X s ,⇡ s ,Q s ,W s )> c 2 for s2 [t,ˆ ⌧ 0 ). Thus, it follows that 2 c 2 E tw {⌧ t} c 2 . When is small, this inequality cannot be true, therefore it leads to a contradiction. Thus, we proved that the supersolution property. We now turn to the visocsity subsolution. Given (t,x,k,q,w), let ' be such that V ' attains its maximum at (t,x,k,q,w)with[V '](t,x,k,q,w)=0. 43 For any> 0, we denote that ⌧ = t+ ^ T t,w 1 .Thisindicatesthattheviscosity subsolution property. Therefore we proved the Theorem. We first observe that I(V,⇡,⌧ ):=E tw n Z ⌧ t U(X s ,Q s )d⇡ s +V(⌧,X ⌧ ,⇡ ⌧ ,Q ⌧ ,W ⌧ ) o V(t,x,k,q,w) 0. Then applying lemma 1.5.1 and sending ! 0, we can derive {V t +L[V]}(t,x,k,q,w) 0. Also we know that V ' attains its maximum at (t,x,k,q,w)with[V '](t,x,k,q,w)=0. Thisindicatesthat[V xx ' xx ](t,x,k,q,w)  0. Then we can claim that {' t +L[']}(t,x,k,q,w){ V t +L[V]}(t,x,k,q,w) 0. (1.4) By the inequality (1.4), we proved the subsolution property. Combine the subso- lution property and supersolution property, we can say our value function V is a viscosity solution to the HJB. 44 1.5.3 Constrained Viscosity Solution We should note that even a classical solution to HJB (1.2) and (1.3) may have discontinuity on the boundary{x=0}[{ w=0}[{ w =T}[{ k=0}[{ q=0}, where HJB (1.3) only specifies the boundary value at t = T and k = K.Inthis section,thenotationofthedomainwilllooktedius,sowewillsimplifythenotation first. Let O := [0,T]⇥ ¯ O t ,O :=intO=(0,t)⇥O t andO ⇤ := [0,T)⇥ ¯ O t . Definition 1.5.4. LetO✓O ⇤ be a subset such that @ T O := {(T,x,k,q,w) 2 @ O6=;}, and let v 2 C(O). We say that v is a constrained viscosity solution on O ⇤ if it’s both a viscosity subsolution on O ⇤ and a viscosity supersolution on O. Now we need to argue why it makes sense that we can extend the subsolution property to O ⇤.Considerthepoint(t,0,k,q,w) 2 @ O ⇤.Let ' 2 C(O)besuch that 0=[V '](t,0,k,q,w)= max ( ˆ t,ˆ x, ˆ k,ˆ q,ˆ w)2O ⇤ ) [V ']( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w). (1.5) Then we must have (@ t ,r )(V ')(t,0,k,q,w)= ↵v. (1.6) 45 where↵> 0, r =(r x ,r k ,r q ,r w ), and v is the outward normal vector at the boundary{x=0},i.e(0, 1,0,0,0). By calculation, we can derive {' t +L[']}(t,0,k,q,w)={V t +L[V]}(t,0,k,q,w)+↵ 0. (1.7) For the boundary{w=0} and{w =t},wecanderivetheoutwardnormalvectors v=(0,0,0,0, 1)andv=( 1,0,0,0,1). Since[V xx ' xx ] 0, wewouldalsolead to (1.7). Similar argument can also be applied to boundary{q=0} and{k=0}. Therefore it’s actually valid to extend subsolution to the boundary ofO. Theorem 1.5.5. The value function is not only a viscosity solution but also a constrained viscosity solution on O ⇤ . Proof. By Theorem 1.5.3, we have already proved that V is a viscosity supersolution. It suces to show that V is also a viscosity subsolution on O ⇤ . Suppose V is not a viscosity subsolution on O ⇤.Thenweclaimthatthere 46 exists (t,x,k,q,w)2O ⇤ , ' 2 C(O), and constants"> 0,⇢> 0, such that 0=[V '](t,x,k,q,w) = max ( ˆ t,ˆ x, ˆ k,ˆ q,ˆ w)2O ⇤ [V ']( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w), but 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : ' t +L['] ( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w) ✏, ( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w)2B ⇢ (t,x,k,q,w)\O ⇤ \{k =K}; V( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w) '( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w) ", ( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w)2@B ⇢ (t,x,k,q,w)\O ⇤ . (1.8) where B ⇢ (t,x,k,q,w)istheopenballcenteredat(t,x,k,q,w)withradius ⇢.To prove this claim, one can note that if V is not a viscosity subsolution onO ⇤ ,then there must exist (t,x,k,q,w)2O ⇤ and ' 0 such that 0 = [V ' 0 ](t,x,k,q,w)= max ( ˆ t,ˆ x, ˆ k,ˆ q,ˆ w)2O ⇤ [V ' 0 ]( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w), but {' 0 t +L[' 0 ](t,x,k,q,w)} = 2⌘< 0 for some ⌘> 0. We need to consider two cases. Case 1.q> 0. We construct '( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w):= ' 0 (t,x,k,q,w)+ ⌘ ⇥ ( ˆ t t) 2 +(ˆ x x) 2 +( ˆ k k) 2 +(ˆ q q) 2 +(ˆ w w) 2 ⇤ 2 (w)(q 2 +w 2 ) 2 . 47 One can easily check that '(t,x,k,q,w)and ' 0 (t,x,k,q,w)havesamegradient and same second order derivative with respect to x.Moreover,ithas (w) Z '(t,x,k,(q+u) + ,0)v(du) (w) Z ' 0 (t,x,k,q,w)v(du)+⌘. Then these lead to ' t +L['] (t,x,k,q,w) ' 0 t +L[' 0 ] (t,x,k,q,w)+⌘< ⌘< 0. Now by continuity of ' t +L['], we can certainly find a⇢> 0 such that 8 > > < > > : ' t +L['] ( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w) ⌘ 2 , ( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w)2B ⇢ (t,x,k,q,w)\O ⇤ \{k =K}. Note for ( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w)2@B ⇢ (t,x,k,q,w)\O ⇤ ,onecanalsoeasilyhas V( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w) '( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w) ⌘⇢ 4 (w)(q 2 +w 2 ) 2 . Then by choosing "=min{⌘, ⌘⇢ 4 (w)(q 2 +w 2 ) 2 },weprovedtheclaim. 48 Case 2. Whenq=0. Wesimplydefine '( ˆ t,ˆ x, ˆ k,ˆ q, ˆ w):= '(t,x,k,q,w)+⌘ ⇥ ( ˆ t t) 2 +(ˆ x x) 2 +( ˆ k k) 2 +(ˆ q) 2 +(ˆ w w) 2 ⇤ 2 and define "=min{⌘,⇢ 2 }.Thus,we proved the claim. Now fix any ⇡ 2A w,k,q ad [t,T], and define ⌧ ⇢ := inf{s>t:(s,X t,x s ,K s ,Q s ,W s ) / 2 B ⇢ (t,x,k,q,w)\O ⇤ }, ⌧ := ⌧ ⇢ ^ (t+T t,w 1 ). Then we can have Z ⌧ t U(X s ,Q s )d⇡ s +V(⌧,X ⌧ ,K ⌧ ,Q ⌧ ,W ⌧ )= Z ⌧ t U(X s ,Q s )d⇡ s +'(⌧,X ⌧ ,K ⌧ ,Q ⌧ ,W ⌧ )+ ⇣ V(⌧,X ⌧ ,K ⌧ ,Q ⌧ W ⌧ ) '(⌧,X ⌧ ,K ⌧ ,Q ⌧ ,W ⌧ ) ⌘ . Observe that on the set {⌧ ⇢ <T t,w 1 } i.e. ⌧ = ⌧ ⇢,and(⌧ ⇢ ,X ⌧ ⇢ ,K ⌧ ⇢ ,Q ⌧ ⇢ ,W ⌧ ⇢ )2 @B ⇢ ^O ⇤ ,wecanhave[V(⌧ ⇢ ,X ⌧ ⇢ ,K ⌧ ⇢ ,Q ⌧ ⇢ ,W ⌧ ⇢ ) '(⌧ ⇢ ,X ⌧ ⇢ ,K ⌧ ⇢ ,Q ⌧ ⇢ ,W ⌧ ⇢ )] ". Then we can have Z ⌧ t U(X s ,Q s )d⇡ s +V(⌧,X ⌧ ,K ⌧ ,Q ⌧ ,W ⌧ ) = Z ⌧ t U(X s ,Q s )d⇡ s +'(⌧,X ⌧ ,K ⌧ ,Q ⌧ ,W ⌧ ) + ⇣ V(⌧,X ⌧ ,K ⌧ ,Q ⌧ W ⌧ ) '(⌧,X ⌧ ,K ⌧ ,Q ⌧ ,W ⌧ ) ⌘  Z ⌧ t U(X s ,Q s )d⇡ s +'(⌧,X ⌧ ,K ⌧ ,Q ⌧ ,W ⌧ ) "1 {⌧ ⇢  t+T t,w 1 } . 49 Before we move forward, we claim that for any s 2 (t,⌧ )where ⌧ = ⌧ ⇢,itholds that H['](s,X s ,⇡ s ,Q s ,W s ) ✏. (1.9) To prove this, we observe that on the set{⌧ ⇢ <t+T t,w 1 },wecanhave ⌧ = ⌧ ⇢.By the definition of ⌧ ⇢,wecanclaimthat(s,X s ,K s ,Q s ,W s )2 B ⇢ (t,x,k,q,w)\O ⇤ , then it proved the (1.9). Furthermore, by applying Ito’s formula like Lemma 1.5.1, E tw n Z ⌧ t U(X s ,Q s )d⇡ s +V(⌧,X ⌧ ,K ⌧ ,Q ⌧ ,W ⌧ ) o  E tw n Z ⌧ t U(X s ,Q s )d⇡ s +'(⌧,X ⌧ ,K ⌧ ,Q ⌧ ,W ⌧ ) "1 {⌧ ⇢  t+T t,w 1 } o = '(t,x,k,q,w)+E tw n Z ⌧ t H['](s,X s ,⇡ s ,Q s ,W s )ds "1 {⌧ ⇢  t+T t,w 1 } o = V(t,x,k,q,w)+E tw n Z ⌧ t H['](s,X s ,⇡ s ,Q s ,W s )ds "1 {⌧ ⇢  t+T t,w 1 } o  V(t,x,k,q,w) "(⌧ t). Since ⇡ is arbitrary, then this contradicts the DPP. Therefore V must satisfy the viscosity subsolution inO ⇤ . 50 Chapter 2 On the Dynamic Frontiers of the Limit Order Books under Equilibrium Model 2.1 Introduction In this chapter, we try to extend dynamic model of LOB proposed in [41] in two major aspects. The guiding idea is to specify the expected equilibrium utility function,whichplaysanessentialroleinthemodelingoftheshapeoftheLOB in that it endogenously determines both the dynamic density of the LOB and its frontier. More precisely, instead of assuming, more or less in an ad hoc manner, that the equilibrium price behaves like an “utility function”, we shall consider it as the consequence of a Bertrand-type game among a large number of liquidity providers (sellers who set limit orders). Following the argument of [51], we first study an N-seller static Bertrand game, each with a profit function involving not only the limit order price less the waiting cost, the same criterion as that in [41], 51 but also the average of the other sellers limit orders observed. We show that the Nash equilibrium exists in such a game. With an easy randomization argument, we can then show that, as N!1 , the Nash equilibrium converges to an optimal strategy of a single player’s optimization problem with a mean-field nature, as expected. We assume there are two types of traders in the market, who only submit market orders (we call it MOs trader) and the other place limit orders (we call it LOs trader). In this paper, we only consider sell side, however, we allow the LOs trader can also submit market orders. In the midprice model, ⌫ denotes the trading speed of LOs trader. There are three fundamental pillars in finance: optimal investment decision problem, pricing and hedging problem, risk measurement and management prob- lem. ThemostfamousfinancialtheoriesrelatedtothefirstpillararetheMarkowitz mean-variance analysis [53] and sharpe’s CAPM model. The most well-known result related to the second pillar is the Black-Scholes-Merton formula. As for the third pillar, the risk measure theory plays an important role, especailly the coherent risk measures. In 1997, Artzner, Delbaen, Eber and Heath [3] introduced the concept of coherent risk measures as a new way of measuring risk. Since then, the theory of coherent risk measures has rapidly been evolving. Readers can find related information in the following papers [[23], [30], [31], [43]]. In this project, we are trying to apply ideas from the first pillar and the third pillar to solve the 52 pricing problem, i.e. We want to propose a model to determine the price of a stock or an asset by applying sharpe ratios and coherent risk measures in a continuous time setting. More precisely, we want to figure out how to determine the dynamic of the best ask/bid price for a stock or a derivative in continuous time. If we want to use the coherent risk measure to construct the NGD bounds, the relevance of the selected pricing density remains an important issue. In most of the papers, researchers apply the equivalent measures as the pricing probability measures. We believe it’s kind of broad to use the equivalent probability set. It’s known that the pricing probability kernels are essentially the beliefs of traders in the market. There are several explanations about these beliefs. Some believe each of these pricing probability measures can be interpreted as a possible model and arbitrage. Or it can be interpreted as di↵erent scenarios of volatilities. And in modern asset pricing theory, people also assume that agents’ beliefs can be interpreted as the likelihood of future states of the world, which is equivalent to beliefs about the excess returns of a security. So when it comes to use the coherent risk measures to price a security, it may not be proper to just simply consider all equivalent probability measures. In our project, we follow the idea that we can interpret the pricing probability measures as traders’ perspective to the market, which is equivalent to beliefs about the excess returns of a security. Now the question is what inputs we should consider in order to apply coherent risk measure to determining the NGD bounds. We follow the idea proposed by 53 Peter Carr and Dilip B.Madan(2000) [15] , Bielecki and Cialenco [11] by using future payo↵ or dividend cash flow as the input. The logic is that If we compare investing stock as investing a project, it will be natural to think about the NPV method from corporate finance to price the asset. In NPV method, we want to choose the project that can maximize the expected utility. So long as an investors’ behavior is consistent with Von Neumann Morgenstern Axioms [56], an investor will accept an opportunity if and only if it increases her expected utility. And normally, we use the project’s future payo↵ as one of the inputs in the NPV method. The central idea in our method is to establish an acceptable set first that every reasonable person would take the view that the benefits engendered by the gains adaqutely compensate for the costs imposed by the losses. We claim a price is accepted if and only if the expected gain under each measure at least weakly exceeds the predetermined constant for each measure in the specified set. This constantcanbecalibratedbythemarketdataandwouldruleouttheacceptability of certain arbitrages. We also recognize that there many self-financing strategies available to the traders. As a result we define an price to be acceptable if it can be financed and hedged so that the expected gain under each measure weakly exceeds its constant, for each probability measure in the specified set. —————— 54 One of the advantages to apply dynamic coherent risk measure is that it has a strong link to Peng’s non-linear expectation theory, also called g-expectation [65]. The other advantage is that by linking to the g-expectation, we may write the best ask/bid prices as the solution to corresponding Backward Stochastic Di↵erential Equations. We proved that for any stock or asset, its dynamic of the best/ask price corresponds to a backward stochastic di↵erential equations. 2.2 Preliminaries Throughout this project we assume that all the market randomness is defined on agivenfilteredprobabilityspace(⌦ ,F,P), on which is defined a d-dimensional standard Brownian motion B = {B t : t 0}.Forconvenience,weshalloften simply assume, unless otherwise specified, that⌦ =⌦ 0 :=C([0,T];R d ), for some giventimehorizonT> 0,F =F 0 :=B(⌦ 0 )istheBorel -fieldon⌦ 0 ,andP =P 0 is the Wiener measure on (⌦ 0 ,F 0 ). We denote the natural filtration generated by the Brownian motion byF B := (F B t ) 0 t T ,andweassumethatF B is augmented so that it satisfies the usual hypotheses. In what follows, we often use ⌦ and ⌦ 0 alternatively, when the context is clear. We denoteL 0 (⌦) to be the space of all random variables defined on (⌦ ,F,P), L p (⌦), p 1, the subspace of all ⇠ 2 L 0 (⌦) such that k⇠ k p p := E[|⇠ | p ] < 1;and L 1 (⌦) the subspace of all ⇠ 2 L 0 (⌦) such that k⇠ k 1 := esssup !2 ⌦ |⇠ (!)| < 1. Furthermore, for a generic Euclidean spaceE,wedenote|·| be its norm, and (·,·) 55 be its inner product. We define L p F ([0,T]⇥ ⌦; E)tobeall E-valued, F-adapted process ⌘ ={⌘ t : t2 [0,T]},suchthat k⌘ k p L p ([0,T]⇥ ⌦; E) :=E ⇥R T 0 |⌘ t | p dt ⇤ <1.We often simply denotek⌘ k L p ([0,T]⇥ ⌦; E) =k⌘ k p as well, when the context is clear. The spaces L 0 F ([0,T]⇥ ⌦; E)and L 1 F ([0,T]⇥ ⌦; E)canthenbedefinedinanobvious way. In particular, if E =R, we shall drop E from the notations for simplicity. We consider an asset in an order-driven market, and focus on its limit order book (LOB). In this project we shall assume that all the buyers are impatient and submit only market orders, but we allow sellers to submit both market and limit orders, and consequently we shall consider only the LOB on the sell side. Unlike our previous work [41] we shall assume that the market orders (both buy and sell) will have a certain price impact, on a macro level, on the fundamental price (or mid price), and we are interested in the collective e↵ect of all the orders on the movement of the frontier, which will in turn a↵ect the “shape’ of the LOB, whence the liquidity cost, etc. (see [41] for the details on these connections). Mathematically, as in [41] we shall assume that the mid-price is a di↵usion process driven by the Brownian motion B.Inparticular,weshallassumethatit has the following general form: dX t =b(t,X t ,µ t ,⌫ t )dt+ (t,X t )dB t ,X 0 =x, t2 [0,T], (2.1) 56 whereb, are deterministic functions µ t is the total number of market buy orders, and ⌫ t is that of the sell orders at time t.Weshouldremarkthatthepriceprocess X could be understood as the return or log price of the asset (or the so-called Bacheliere model), hence less sensitive to the positiveness of the process. This will be made more rigorous in our formal set-up later. We shall make use of the following standing assumptions on the coecients. Assumption 2.2.1. The functions b:[0,T]⇥ R 3 7! R and :[0,T]⇥ R 7! R enjoy the following properties: (i) There existsK> 0, such that for x,y 2 R and (t,µ,⌫ ) 2 [0,T]⇥ R 2 , it holds that |b(t,x,µ,⌫ ) b(t,y,µ,⌫ )|+| (t,x,) (t,y)| K|x y|; (ii) The function b is increasing in µ, and decreasing in ⌫ . Remark 2.2.2. Clearly, under Assumption 2.2.1, for any given processesµ,⌫ ,the SDE (2.1) is well-posed, and by the comparison theorems for SDEs, we know that the mid-priceX =X µ,⌫ will be “pushed” up by the buy ordersµ,anddownbythe sell orders ⌫ .Weshouldnotethatsuchamodelisinthesamespiritoftheones proposed in [17], in which b(t,x,µ,⌫ )=g(µ) g(⌫ ) for some linear, non-negative, increasing function g,and (t,x)⌘ is a constant.In other words, our model is a simple generalization of that of [17] in a nonlinear form. 57 It is worth noting that, as part of the strategy of the (collective) seller, the process ⌫ will also play a role in the determination of the frontier. We shall therefore consider it as a control process.Wewillgiveamorespecificdescription of the so-called admissible strategy set in §7, as a subspace ofL 2 F ([0,T]⇥ ⌦). Throughout this paper we assume that there is a common discounting factor in the market, denoted by ={ T t } t 0 . Also, we shall assume that there is a cash flow process of the firm (i.e., the issuer of the stock) that is commonly observable by all the sellers, denoted by ⇠ = {⇠ t } t 0 . Without specifying the interest rate or dividend rate, we shall simply assume that the process T and ⇠ satisfy the following natural conditions: Assumption 2.2.3. (i) For each t2 [0,T], the process t 2L 1 ([0,t]⇥ R), such that s7! t s is decreasing, t s > 0, s2 [0,t], P-a.s., and t t = s s =1; (ii) The process ⇠ 2L 2 F ([0,T]⇥ ⌦) , such that t7! ⇠ t is decreasing, and ⇠ t > 0, t2 [0,T], P-a.s. Remark 2.2.4. It is not unusual to assume that the cash flow process ⇠ is the cumulated dividend process, whence increasing. We should note that the dis- counted cash flow of the firm, T t ⇠ T , t2 [0,T], is often used as the present value of the potential future payo↵ at time t to determine the so-called No Good Deal bound (cf. e.g., [11], [15]). In this paper we shall use it to present the potential 58 loss for the seller when they decide the position of the sell limit order (see §4for the detailed discussion). To end this section we shall briefly describe the market where all the sellers can make investments. In particular, we assume that, besides the the specific stock that the seller is looking to sell, there are another N risky assets and one riskless asset in the market that the seller can access, and the limit order that the seller choosestosetisamongthepositionsthatthesellercan“superhedge”thepotential loss for giving up the cash flow that is generated by holding the stock (see §4for details). At this point we do not specify the dynamics of the price processes for the N risky assets (which will conceivably depend on each seller’s perspective towards the market, specified by an associated probability measure, see §3below). Atthis pointweshallonlyassumethattheprices, denotedbydenotedbyS :={S i } 1 i N , are nonnegative, locally boundedF-semimartingales, under each seller’s subjective measure. With a slight abuse of notaion, we shall denoteL(S)tobethespaceof all F-predictable N-dimensional processes that are integrable with respect to S, under the associated probability measure. Atradingstrategy,orportfolio,ofaparticularinvestor(aseller,inourcase) is a vector valued F-predictable process H=(H 1 ,..,H N )2L(S), where for each 0  i  N and t 2 [0,T], H i t denotes the number of shares of i-th asset held at time t,for i=1,···,N. Now, denoting H 0 t to be the amount of money in the 59 money market account at time t,foragiventradingstrategy(H,H 0 ), we denote it’s value by ¯ V H t = P N i=1 H i t S i t +H 0 t , t 0. In this paper, we shall also assume the presence of the transaction cost,whichcouldcomefromeitherastransactionfees or fixed commissions, or the so-called “waiting cost” that is common in the case of LOB. We shall assume (see, e.g., [41], ) that the discounted cumulated transaction cost over [t,T]✓ [0,T]takestheform C H,t s = N X i=0 Z s t r t c i r dr, s2 [t,T]. (2.2) where c i t 0, i=1,···,N,isthetransactioncostincurredbytradingthe i-th asset and c 0 t can be thought of as the waiting cost. For simplicity we shall assume that c i ’s are uniformly bounded. Now, assume that the trading strategyH is self-financing,thenthediscounted value of H,denotedby V H t = t ¯ V H t , t2 [0,T], should have the dynamics V H t =V H 0 + Z t 0 ( s H s ,dS s ) C H t =V H 0 + N X i=1 Z t 0 t H i s dS i s C H t ,t2 [0,T].(2.3) We shall often focus on the following admissible self-financing strategies. Definition 2.2.5. A self-financing trading strategy (H,H 0 ) on [t,T], is called “admissible” if: (i) V H 0 =0, 60 (ii) V H T = P N i=1 R T 0 t H i r dS i r C H,t T c, for some c 0; (iv) H i t 0, t2 [0,T], P-a.s., for alli>d. We denoteH to be the set of all admissible self-financing trading strategies. Remark 2.2.6. a) We should note that each conditions in Definition 2.2.5 holds “almostsurely”,withrespecttothesubjectiveprobabilitymeasureassociatedwith each seller. However, when we consider all the sellers, especially in the case when themarketisincomplete,weneedtoutilizethenotionoftheso-called“quasi-sure”, which will be described in the next section. b) The condition (iii) in Definition 2.2.5 amounts to saying that only the first d (d> 0) assets (including the money market account) can be sold short in an admissible strategy, whereas the restN d risky assets cannot be sold short under any circumstances. 2.3 The Sellers The main idea of our description of the limit order book (whence its frontier) is the following micro-view of the liquidity providers, that is, all the sellers who provide the (sell) limit orders. Our basic assumption is that di↵erent seller has di↵erent view towards the market, due to, say, the di↵erent information the seller possesses regarding the market parameters. Mathematically, we shall assume that each seller is represented by a (subjective) probability measure (or a “prior”) on 61 the canonical space (⌦ 0 ,F 0 ), and denote the totally of all the sellers subjective measures byP := {P ✓ : ✓ 2 ⇥ },where⇥issomeindexset,whichisassumed to be countable, for technical convenience. We note that at this point we do not impose any constraints on the setP,thereforetheelementsinP could very well be mutually singular. Let us now add some structure to the setP,borrowingsomeideaof model uncertainty or ambiguity,inparticularthatof[27](or[7]). Tobemoreprecise, let us label each seller by a parameter “✓ ”, choosing from an index set ⇥. We assume that each seller is allowed to invest in a market that contains N assets, as was mentioned in §2. However, we shall now assume that each seller will use his/her own “prior” to evaluate the assess, in the sense that they are allow to have their own market parameters. In other words, we assume that each priorP ✓ 2P is actually the law of the solution to an N-dimensional SDE under P 0 ,thatis, P ✓ =P 0 (X ✓ ) 1 ,and dX ✓ t =b ✓ (t,X ✓ ·^ t )dt+ ✓ (t,X ✓ ·^ t )dB ✓ t ,X ✓ 0 =x 0 ,t2 [0,T], (2.1) where b ✓ and ✓ are progressively measurable functional defined on [0,T] ⇥ C([0,T];R N ),representingtheappreciationrateandvolatilitymatrix,respectively, and B ✓ is aP ✓ -Brownian motion. We note here that the process X ✓ can be inter- preted as the “return” (or the “log-price”) of the assets from seller ✓ ’s perspective, 62 so that the actual price process satisfies the standard (generalized) Black-Scholes SDE: dS ✓ t =diag[S ✓ t ]dX ✓ t =diag[S ✓ t ][b ✓ (t,·)dt+ ✓ (t,·)dB ✓ t ],S ✓ 0 =s 0 ,t2 [0,T]. (2.2) Here the matrix diag[S ✓ ]denotesthediagonalmatrixwithdiagonalentriesbeing S ✓ . Furthermore, in what follows we shall also assume that seller ✓ also has its own view of the risk-free interest rate due to, for example, his/her own term structure model, or other reasons of parameter uncertainty, and denote it by r ✓.Given such a specification, in what follows we shall follow the convention (see, e.g., [27]) and simply identify ✓ ⇠ (b ✓ , ✓ ,r ✓ ). In other words, we shall identify the index set⇥ as the totality of all possible choices of the market parameters from sellers’ perspectives. Clearly, there is no particular reason at this point to assume that the measures P ✓ ’s are absolutely continuous to each other. We shall, however, make use of the following standing assumptions for the market parameters in the rest of the paper: Assumption 2.3.1. (i) d =N; 63 (ii) The interest rates r ✓ = {r ✓ t } t2 [0,T] , ✓ 2 ⇥ , r = {r t } t2 [0,T] , and the trans- action costs c i ={c i t } t2 [0,T] , i=1,···,N, are positive, F-progressively measurable processes. Furthermore, there exist constants K,K ✓ > 0, ✓ 2⇥ , such that |r ✓ t | K ✓ , |r t |+max i |c i t | K, t2 [0,T], P 0 a.s. (iii) The appreciation-volatility pairs (b ✓ , ✓ ), are F-progressively measurable processes with appropriate dimensions, such that for some  2 L p F ([0,T]),p> 2, it holds that |b ✓ t |+| ✓ t |< t ,dt⌦ dP 0 a.e., ✓ 2⇥; (2.3) (iv) There exist 0 > 0 andK> 0, such that 8 > > > < > > > : ✓ t ( ✓ t ) T 0 I d , |[ ✓ t ] 1 b ✓ t | K, dt⌦ dP 0 a.e., ✓ 2⇥ , (2.4) where I d is the d⇥ d identity matrix; Remark2.3.2. (a)Weshouldremarkthatthed-dimensionalSDE(2.2)represents the asset princes of the d (=N)assetsthateachsellercaninveston,withhis/her 64 own evaluation of the parameters (r ✓ ,b ✓ , ✓ ). This lack of an unified evaluation of the market parameters, together with the transaction cost to be introduced later, makes the market naturally incomplete. (b)TheAssumption2.3.1canalsobeinterpretedintermsofambiguityonboth the risk return and volatility, in the sense of, say, [7,27]. In particular, we shall assumethat⇥containsallelements( r,r, ), wherer ={r t },thetrueinterestrate, and is any constant matrix satisfying Assumption 2.3.1. (c) The assumption that all the processes r ✓ ,r,c i are bounded is merely tech- nical, so as to facilitate the arguments when the Girsanov theorem is applied. But it is worth noting that we do not assume that r ✓ ’s are uniformly bounded. That is, sup ✓ K ✓ =1 is possible. Under Assumption 2.3.1, we have the following simple result, which we give for ready reference. Proposition 2.3.3. Assume that Assumption 2.3.1 is in force. Then the setP is weakly pre-compact in the sense of probability measures. Proof. Since X ✓ ’s all have continuous paths, in light of [37, Theorem 3.9] (see also [36]), we need only show that, for someq> 0, and`> 1, it holds for all ✓ 2⇥ and 00dependingonlyonT> 0. To this end, let us recall Assumption 2.3.1-(iii), and denote C T >0tobea generic constant that depends only onT>0andisallowedtovaryfromlineto line. For anyq> 0, 0 2p p 2 we see that (2.6) leads to (2.5), proving the proposition. We now give a more detailed characterization of the set of probabilitiesP, whichwillbeimportantforourdiscussionbelow. Weshallborrowsomearguments in the spirit of the weak formulation of the stochastic optimization problems and market with uncertainties (see, e.g., [7,67] or the book [78]). To begin with, we recall the canonical probability space (⌦ ,F,P 0 )andthecanonicalBrownian motionB. Consider the set of probability measures defined on the canonical space (⌦ ,F): P 0 :={P ,✓ :=P 0 (X ✓ ) 1 :X ✓ · := Z · 0 ✓ (s,X ✓ ·^ s )dB s , P 0 a.s., ✓ 2⇥ }. (2.7) 66 WenotethatunderAssumption2.3.1,theSDEin(2.7)hasauniquestrongsolution for each given ✓ ,andthefiltrationgeneratedby X ✓ is the same as the original (Brownian) filtration F = {F t }.Furthermore,undertheprobability P ,✓ ,the canonical process is now X ✓ . Namely, X ✓ t (!)= !(t), forP ,✓ -a.e. !2⌦. We can therefore rewrite ✓ (t,X ✓ ·^ t (!)) := ✓ t (!),t2 [0,T], P ,✓ a.s. (2.8) Since underP ,✓ , X ✓ is a continuous martingale, with quadratic variation pro- cess hX ✓ i t = R t 0 ˆ a ✓ s ds,whereˆ a ✓ t := ✓ t ( ✓ t ) T := [b ✓ t ] 2 ,andb ✓ t := [ ✓ t ( ✓ t ) T ] 1/2 takes values in allN⇥ N positivelydefinitesymmetricmatrices. Therefore, theprocesss B ,✓ t := Z t 0 [b ✓ s ] 1 dX ✓ s ,t2 [0,T] (2.9) is aP ,✓ -Brownian motion. In other words, underP ,✓ we can write dX ✓ =b ✓ t dB ,✓ t ,t2 [0,T]. (2.10) 67 Furthermore, by Girsanov theorem, for each ✓ we can define a new probability measure b P ✓ that is risk neutral for the seller ✓,inthesensethatthecanonical process X ✓ has the following dynamics: dX ✓ t =r ✓ t dt+b ✓ t dB ✓ t ,X ✓ 0 =x 0 ,t2 [0,T], (2.11) where B ✓ t = B ,✓ t R t 0 [b ✓ s ] 1 r ✓ s ds is a b P ✓ -Brownian motion. Since the seller ✓ will use b P ✓ for his/her risk neutral evaluation of any contingent claim, we shall refer to the probability b P ✓ as the pricing measures for seller ✓.Inotherwords,denoting b ✓ t := [b ✓ t ] 1 r ✓ t , t2 [0,T], and for a (bounded)F-progressively measurable process ={ t },let M t (,B ):=exp n Z t 0 s dB t 1 2 Z T 0 | t | 2 dt o ,t2 [0,T], (2.12) the set of pricing measures can be written as c P := n b P ✓ : d b P ✓ dP ,✓ F T =M T ( b ✓ ,B ,✓ ), P ,✓ 2P 0 o . (2.13) Remark2.3.4. We observe that under each element b P ✓ 2 c P thecanonicalprocess X ✓ hasdynamics(2.11),thus b P ✓ isessentiallythesameasP ✓ =P 0 (X ✓ ) 1 ,defined via(2.1)exceptforthe“riskneutral”requirement(i.e,b ✓ =r ✓ ). Thereforeinwhat 68 follows we shall focus only on the sets c P, and simply denote it byP without further specification. We should note that, despite the explicit forms of the elements, the probability measures inP(= c P)couldverywellbemutuallysingular,sincetheelementsin P 0 are often mutually singular. However, all the probabilities “deviation” from “true” risk neutral one, namely the one corresponding to r ✓ ⌘ r,inanabsolutely continuous manner, determined by a “risk premium” process ✓ t := [b ✓ t ] 1 (r ✓ t r t ), t2 [0,T]. To see this, let us consider a subset c P 0 ⇢ c P,definedby c P 0 := n b P 0,✓ : d b P 0,✓ dP ,✓ F T =M T ( b 0,✓ ,B ,✓ ), P ,✓ 2P 0 o . (2.14) where b 0,✓ =[b ✓ t ] 1 r t , t2 [0,T]. Next, for any b P ✓ 2 c P,werewrite(2.11)as dX ✓ =r t dt+b ✓ t d ˜ B ✓ t ,t2 [0,T], (2.15) where ˜ B ✓ t = R t 0 ✓ s ds+B ✓ t ,t2 [0,T], is a Brownian motion under a new probability measureQ ✓ defined by dQ ✓ d b P ✓ F T =exp n Z T 0 ✓ s dB ✓ s 1 2 Z T 0 | ✓ s | 2 ds o =:M T ( ✓ ,B ✓ )=[M T ( ✓ , ˜ B ✓ )] 1 . 69 But on the other hand, we note that under each b P 0,✓ 2 c P 0 , the canonical (return) process X ✓ has the dynamics dX ✓ t =r t dt+ ✓ (t,X ✓ ·^ t )dB t =r t dt+b ✓ t dB ,✓ t ,t2 [0,T]. (2.16) By the uniqueness in law of the SDE we conclude that Q ✓ (X ✓ , ˜ B ✓ ) 1 = b P 0,✓ (X ✓ ,B ,✓ ) 1 . Thus, with a slight abuse of notation, we can identifyQ ✓ ⇠ b P 0,✓ ,and ˜ B ✓ ⇠ B ,✓ .Conseqently,wecanwrite dQ ✓ d b P ✓ F T =M T ( ✓ ,B ✓ )=[M T ( ✓ , ˜ B ✓ )] 1 =[M T ( ✓ ,B ,✓ )] 1 = h d b P ✓ d b P 0,✓ F T i 1 . Consequently, we can write the setP(= c P)inthefollowingexplicitform: P := n b P ✓ : d b P ✓ d b P 0,✓ F T =M T ( ✓ ,B ,✓ ); b P 0,✓ 2 c P 0 o . (2.17) To end this section we recall some useful notions relevant to our framework. First, we note that the probability measures in the set P are not necessarily mutually equivalent, due to the presence of the “volatility” coecients ✓.In fact, even in the case when r ✓ t ⌘ r t ,forall ✓ 2 ⇥, the laws of X ✓ ’s can still be mutually singular. Therefore in general we should use the generalized notion of the arbitrage opportunity in terms of the so-called “quasi-sure” with respect to the 70 family of probability, initiated by Denis and Martini [25] and Peng ([63,64])(see, e.g., [14]). Let us first recall that a set A2F 0 is called a polar set if P ✓ (A)=0forall P ✓ 2P. We say a property holds “quasi-surely” (q.s.) if it holds outside a polar set, or equivalently, it holdsP-a.s. for allP2P.Wenextstrengthenthenotion of “admissibility” given by Definition 2.2.5 slightly: Definition 2.3.5. A set of self-financing strategiesH is called “strictly adimssi- ble” if it is admissible in the sense of Definition 2.2.5 with respect to everyP ✓ 2P. Next, for any H 2 H,and t 2 [0,T], we denote V ✓,H t , t 2 [0,T], to be the discounted value of H over [0,T]fromseller ✓ ’s perspective, and specify the discounting factor by seller ✓ as ✓ t = e R t 0 r ✓ s ds , t2 [0,T]. The following notion of arbitrage opportunity can be found in [14]. Definition 2.3.6. A (strictly) admissible self-financing strategy H 2H is called an arbitrage opportunity, if (i) for every P ✓ 2P, it holds that that V ✓,H T 0, P ✓ -almost surely; and (ii) there exists at least one ✓ 2⇥ , it holds that E ✓ [V ✓,H T ]:=E P ✓ [V ✓,H T ]> 0. 71 Remark 2.3.7. In the rest of the paper, for notational convenience we make the convention that, under the pricing measureP ✓ for the seller ✓ 2⇥, we shall always use the discounting factor ✓ t =e R t 0 r ✓ s ds .Inotherwords,wewillsimplydenote E ✓ [ T ⇠ T ]=E ✓ [ ✓ T ⇠ T ], E ✓ [V H T ]=E ✓ [V ✓,H, T ], ··· etc., when there is no danger of confusion. 2.4 StructureofDynamicalLOBanditsFrontier Inthissectionweshallexplainthetheoreticalframeworkwechoosetodescribethe dynamical(sell)LOBanditsfrontier,thatis,thebestaskingpriceateachtimet2 [0,T]. Our starting point will be in the spirit of the so-called Good Deal Valuation (GDV) and the Good Deal Bound (GDB) of asset pricing in an incomplete market, which usually refer to a tighter interval for possible risk-neutral prices as opposed to the super-replicating bounds one using “no-arbitrage” assumption. We should note, however, that our definition of a “good deal” is slightly di↵erent from the existing ones (see, e.g., [1,5,12,13,30,42] and the references cited therein), but we still use the same term for lack of a better name. We shall show that our definition will nevertheless lead to the more or less same GDB. Roughly speaking, if an “arbitrage opportunity” is an extremely good deal for investors that should be banned by a reasonable market, a “good deal” usually 72 means to be a weaker, but still unfair advantage that none of the seller should be allowed to have. In this spirit, we shall make use of the following definition. Definition 2.4.1. We say that an admissible self-financing strategy H 2H is a “good deal” if it satisfies the condition (ii) of Definition 2.3.6. That is, there exists ✓ 2⇥ , such that E ✓ [V H T ]> 0. (2.1) Consequently, we say that an LOB observes “No Good Deal” (NGD) rule if for all H 2H , it holds that sup P ✓ 2 P E ✓ [V H T ] 0. (2.2) Remark 2.4.2. (i) By definition, an arbitrage opportunity must be a good deal, but converse if not true. Therefore a No Good Deal restriction would produce a tighter bound than “no arbitrage”. (ii) As we shall see later, this definition of good deal is in fact more closely related to the so-called “relevant good deal” in the literature (see, e.g., [1,24]). We purposely blur the line here, as this is not the main focus of this paper. 73 2.4.1 Acceptable limit orders In this subsection we describe how the sellers are setting their limit orders. To avoid the issue of risk taking tendencies, in this paper we assume all the sellers are “rational” or “risk averse”, in the sense that they will only sell the stock at the price where they can use it to hedge the loss of the dividend income, based on their perspectives towards the market. More precisely, assume that the seller ✓ 2⇥setsalimit(sell)orderataprice x at t=0,andthenchoosea(strictly) admissible strategy H 2H,whichwouldgenerateacashflowwhosediscounted present value is x+V ✓,H t , t2 [0,T]. On the other hand, denote {⇠ t ,t2 [0,T]} to be the dividend plan of the stock, which will be the loss to the seller by selling the stock. The following definition is natural: Definition2.4.3. Given the future dividend plan ⇠ , a sell pricex is called “accept- able” for the seller ✓ 2⇥ if there exists H 2H , such that E ✓ [x+V H T T ⇠ T ] 0, P a.s. (2.3) Here in the above P ✓ 2P is seller ✓ ’s perspective risk neutral measure. Further- more, we define the set of all acceptable sell prices of seller ✓ 2⇥ by S ✓ :={x :9H 2H , E ✓ [x+V H T T ⇠ T ] 0}. (2.4) 74 We begin by looking at tall the possible prices in an LOB. A simple minded definition of such a collection of prices at any time t2 [0,T]couldbe ¯ S := [ ✓ 2 ⇥ S ✓ , and then define the best ask price by ¯ p := inf ¯ S.Weshallarguethatthisisnot reasonable. To see this, note that the definition (2.4) indicates that ifx2S ✓ ,then any y x would also be in S ✓ .Thatis,[x,1)⇢S ✓ whenever x2S ✓ . Also, by choosing H ⌘ 0oneconcludesthatE ✓ [ T ⇠ ]2S ✓ ,andtherefore [E ✓ [ T ⇠ ],1)⇢S ✓ , for all ✓ 2⇥. (2.5) Thisclearlyleadsto ¯ p=inf ¯ S E ✓ [ 0 T ⇠ ], forall ✓ 2⇥. NowrecalltheAssumption 2.3.1-(iv), and consider the “risk neutral” seller ✓ 0 2 ⇥. Then in particular we should have ¯ p  E ✓ 0 [ T ⇠ ], the risk neutral price. But on the other hand, it is obvious that no one would give up a stock for less than the risk neutral present value of a given dividend gain, we must have ¯ p E ✓ 0 [ T ⇠ ]. That is, ¯ p =E ✓ 0 [ T ⇠ ]. But since this essentially says that there is no bid-ask spread, we do not believe the characterization is sensible. 75 Keeping this in mind, we shall argue that the reasonable definition of an LOB should actually be the following set: S := \ ✓ 2 ⇥ S ✓ . (2.6) That is, the set of prices that are acceptable to all the sellers, and consequently the dynamic frontier of the LOB, namely the process of best ask price,shouldbe defined by P ask := inf{x :x2S}. (2.7) Remark 2.4.4.Aheuristicjustificationofdefinitions(2.6)and(2.7)canbeas follows. If a (sell) price is not “acceptable” for a particular seller ✓ 2⇥, then this seller would either hold the stock or buy consider it a good “buy” deal, and would buy it with a market order. Thus the price would not exist in the LOB. In the rest of the section we shall verity that such a definition actually leads to theGoodDealBoundcorrespondingtothestandarddynamicriskmeasuredefined as the upper expectation over the set of probabilitiesP along the lines of, e.g., [1,12], etc. To this end, we first prove an equivalent characterization of the family of setsS ={S t } t2 [0,T] under the No Good Deal assumption. 76 Proposition 2.4.5. Assume that Assumption 2.3.1 is in force. Then, the set S defined by (2.6) can be written in the following form: S ={x2R + :9H 2H such that inf P ✓ 2 P E ✓ [x+V H T T ⇠ T ] 0}. (2.8) Proof. Denote the right hand side of (2.4.5) by e S.Wefirstshowthat e S⇢S . In fact, if x2 e S,thenbydefinitionthereexistsa H 2H,suchthat inf P ✓ 2 P E ✓ [x+V H T T ⇠ T ] 0. Clearly, this implies that x2S ✓ ,foreach ✓ 2⇥. That is, x2\ ✓ 2 ⇥ S ✓ =S. Conversely, suppose that x2S =\ ✓ 2 ⇥ S ✓ .Thenforeach ✓ 2⇥, there exists an admissible trading strategy H ✓ 2H,suchthat E P ✓ [x+V H ✓ T T ⇠ T ] 0. (2.9) Now by Assumption 2.3.1, each H ✓ must satisfy the “no good deal” condition, that is,E ✓ [V H ✓ T ] 0. Thus (2.9) yields thatE ✓ [x 0 T ⇠ T ] E ✓ [V H ✓ T ] 0, for all P ✓ 2P.Consequently,choosing H 0 ⌘ 02H,wehave inf P ✓ 2 P E ✓ [x+V 0 T T ⇠ T ]= inf P ✓ 2 P E ✓ [x T ⇠ T ] 0. (2.10) 77 That is, x2 e S,provingtheproposition. 2.4.2 Frontier and the Good Deal Bound In this subsection we study the relationship between the best ask price defined by (2.7) is the so-called Good Deal Bound in the sense of [1] and [24]. To begin with, we define, for any X 2L 2 F T (⌦), ⇢ (X):= sup P ✓ 2 P E ✓ [ T X]= inf P ✓ 2 P E ✓ [ T X]. (2.11) Here we emphasize the convention that under P ✓ the present value is calculated di↵erently for each seller ✓ 2 ⇥, with discounting factor t = ✓ t = e R t 0 r ✓ s ds , t 2 [0,T]. However one can easily check that ⇢ (·)stilldefinesa coherent risk measure in the sense of, say, [23]. Moreover, by definition (2.7) and Proposition 2.4.5, we see that the best ask price can be written as P ask =infS=inf{x :9H 2H , inf P ✓ 2 P E P ✓ [x+ T ( ¯ V H T ⇠ T )] 0} (2.12) =inf{x :9H 2H,x ⇢ ( ¯ V H T ⇠ T ) 0}, where ¯ V H t =(H T ,S T )+H 0 T is the un-discounted value of the portfolio (H,H 0 ). We have the following duality result. 78 Theorem 2.4.6. Assume that Assumption 2.3.1 is in force. Then, for any divi- dend plan ⇠ , the frontier of the (sell) LOB, or the best ask price is given by P ask =sup P ✓ 2 P E ✓ [ T ⇠ T ]= inf H2 H ⇢ ( ¯ V H T ⇠ T ). (2.13) Proof. For each t 2 [0,T], let us denote p ⇤ := sup P ✓ 2 P E P ✓ [ T ⇠ T ], and ¯ p := inf H2 H ⇢ ( ¯ V H T ⇠ T ). Then (2.13) amounts to saying that P ask = p ⇤ =¯ p,andwe shall proceed by proving the following claims. (i) p ⇤ P ask .Indeed,inlightof(2.5),wehave p ⇤ =sup P ✓ 2 P E P ✓ [ T ⇠ T ]2 [E P ✓ [ T ⇠ T ],1)⇢S ✓ , for all ✓ 2⇥ , That is, p ⇤ 2\ ✓ S ✓ =S,andhence p ⇤ infS =P ask . (ii) P ask ¯ p.Toseethis,notethatforany x2S,by(2.12),thereexists H ⇤ 2H,suchthat x ⇢ ( ¯ V H ⇤ T ⇠ T )=x+ inf P ✓ 2 P E P [V H ⇤ T ✓ T ⇠ T ] 0. Thus x ⇢ ( ¯ V H ⇤ T ⇠ T ) inf H2 H ⇢ ( ¯ V H T ⇠ T )= ¯ p.Since x2S is arbitrary, we have P ask ¯ p. 79 (iii) ¯ p p ⇤ . Again, by virtue of the NGD condition (2.2), we see that, for H 2 H,onehas E P ✓ [V H T ]= E P ✓ [ T ¯ V H T ]  0, for all ✓ 2 ⇥. Therefore, by definition (2.11) we have, for all H 2H , ⇢ ( ¯ V H T ⇠ T )= sup P ✓ 2 P E P ✓ [ ✓ T (⇠ T ¯ V H T )] sup P ✓ 2 P E P ✓ [ ✓ T ⇠ T ]=p ⇤ . (2.14) Consequently, we have ¯ p=inf H2 H ⇢ ( ¯ V H T ⇠ T ) p ⇤ .Thiscompletestheproof. Remark 2.4.7. We note that all the arguments above can be extended to a dynamic manner, in the sense that the frontier of the LOB is determined at each time t2 [0,T]. However, we will not pursue such a generality here as this is not the main purpose of this paper. 2.5 TheConnectiontotheAcceptableIndexand BSDEs In this section, we shall justify the representation of the frontier of the LOB estab- lished in the last section by proving a duality relationship, that is, determine the frontier using the so-called acceptable index introduced in [18] (see also [52] and [11]). To simplify presentation, in what follows we shall assume that all the seller have the same view on the market volatility, namely, we assume that ✓ ⌘ for 80 all ✓ 2 ⇥, and satisfies Assumption 2.3.1. We note that in this case the set P 0 defined by (2.7) becomes a singleton: P ,✓ ⌘ P 0 (X ) 1 =: P ,where X is (strong) solution to the SDE: dX t = (t,X ·^ t )dB t .Clearly,inthiscasewe can simply identify the P -Brownian motion B ,✓ (see (2.9)) with the canonical Brownian motion B,andconsequently,theset c P 0 defined by (2.13) is also a singleton, given by the probability b P ,satisfying d b P dP F T =M T ( ˆ 0 ,B), ˆ 0 =[ t ] 1 r t 1. Therefore we can write the set of pricing measures (2.13) as P := n P ✓ : dP ✓ d b P F T =M T ( ✓ ,B), ✓ 2⇥ o , (2.1) where M t ( ✓ ,B)=exp n R t 0 ✓ t dB t 1 2 R T 0 | ✓ t | 2 dt o ,and ✓ t = 1 t (r ✓ t r)1, t 2 [0,T]. Note that Assumption 2.3.1-(ii) guarantees that each M( ✓ ,B)isa b P -martingale, and underP ✓ , B ✓ t = R t 0 ✓ s ds+B t is a Brownian motion. To establish the connection to the so-called acceptable index, let us denote, for K> 0, P K := n P ✓ : dP ✓ d b P F T =M T ( ✓ ,B), | ✓ | K, ✓ 2⇥ o . (2.2) 81 Then it is readily seen that the family {P K } K>0 is increasing in K,andP = lim K!1 P K =[ K>0 P K .Furthermore,similarto(2.11)wecandefine,for X 2 L 2 F T (⌦) and K> 0, ⇢ K (X):= sup P ✓ 2 P K E ✓ [ T X]= inf P ✓ 2 P K E ✓ [ T X]. (2.3) Remark 2.5.1. Since {P K } K>0 is an increasing family of pre-compact sets, for each X 2L 2 F T (⌦), the mapping K 7! ⇢ K (X) is increasing as well. It is not hard to check that it is left-continuous, that is, lim %K ⇢ (X)= ⇢ K (X). Moreover, in light of (2.12) and Theorem 2.4.6, for eachK> 0, we define P ask,K := inf{x :9H 2H,x ⇢ K ( ¯ V H T ⇠ T ) 0}=inf H2 H ⇢ K ( ¯ V H T ⇠ T ) =sup P ✓ 2 P K E ✓ [ T ⇠ T ]. (2.4) We shall refer to P ask,K as the “K-level frontier” of the (sell) LOB for obvious reasons. Let us now recall the notion of acceptable index (see, for example, [18] [52] or [11]). Definition 2.5.2. A mapping ↵ :L 2 F T (⌦) ! R + is called an acceptable index (AI) if it satisfies the following properties: 82 (i) (Quasi-concavity) If ↵ (⇠ 1 ) c and ↵ (⇠ 2 ) c for some constant c 0, then ↵ (⇠ 1 +(1 )⇠ 2 ) c, 2 (0,1); (ii) (Monotonicity) If ↵ (⇠ 1 ) c for some c 0 and ⇠ 2 ⇠ 1 , then ↵ (⇠ 2 ) c; (iii) (Fatou property) For any sequence {⇠ n } n2 N such that |⇠ n | 1 and ⇠ n ⇠ in probability, if ↵ (⇠ n ) c for some c 0, then also ↵ (⇠ ) c; (iv) (Scale invariant) ↵ (⇠ )= ↵ (⇠ ) for any> 0. In particular, if we define, for X 2L 2 F T (⌦), and x> 0, ↵ x (X):=sup{K>0: inf P ✓ 2 P K E ✓ [ T 0 X] x}=sup{K>0: ⇢ K (X) x}. (2.5) Then it can be shown (see, e.g., [18, Theorem 1]), ↵ (·)isanacceptableindex,and similar to [11, Lemma B.1] (see also [52]), we have the following result. Theorem2.5.3. Assume Assumption 2.3.1 holds and assume further that ✓ ⌘ , ✓ 2 ⇥ . Then, for eachK> 0, the “K-level” frontier of the (sell) LOB, P ask,K defined by (2.4), can be expressed in terms of the acceptable index in the following way: P ask,K := inf{x>0:9H2H,↵ x ( ¯ V H T ⇠ T ) K}. (2.6) Furthermore, it holds that P ask =lim K!1 P ask,K . 83 Proof First recall from Remark 2.5.1 that the mappingK 7! ⇢ K (X)isincreas- ing, and left-continuous. We claim that for any ⇠ 2L 2 F T and x,K > 0, ⇢ K (⇠ ) x if and only if sup{ : ⇢ (⇠ ) x} K. (2.7) Indeed, fixingx> 0anddenoting A x := { : ⇢ (⇠ )  x},then ⇢ K (⇠ )  x implies that K 2A x ,hencesupA x K.Conversely,denote x := supA x and assume K  x < 1 (as x = 1 is trivial). By definition of “sup” we can find asequence n 2A x and n % x .Sinceforeach n, ⇢ n (⇠ ) x and 7! ⇢ (⇠ )is left-continuous, we have ⇢ x (⇠ )=lim n% x⇢ n (⇠ )  x.Themonotonicityofthe mapping 7! ⇢ (⇠ )thenimpliesthat ⇢ K (⇠ ) ⇢ x (⇠ ) x,proving(2.7). Now by definitions (2.4), (2.5) and the fact (2.7) we have P ask,K =inf{x :9H,x+inf P2 P K E P [V H T T 0 ⇠ T ] 0} =inf{x :9H,x ⇢ K ( ¯ V H T ⇠ T ) 0} =inf{x :9H,sup{> 0: ⇢ ( ¯ V H T ⇠ T ) x} K} =inf{x :9H,↵ x ( ¯ V H T ⇠ T ) K}. Therefore the best ask price in our model is essentially equivalent to the best ask price induced by an acceptable index. 84 To end this section, we give a representation of the frontier via the Backward Stochastic Di↵erential Equation (BSDE). We should note that the representation is due largely to the assumption ✓ ⌘ that we have been using throughout this section. In the general case when ✓ is allowed to vary, the notion of “second order BSDE” (see, e.g., [7]) would be a proper tool, but we will not go into the details as this is not the main purpose of this paper. The idea is more or less standard. For any given dividend plan ⇠ ={⇠ t } t2 [0,T] and ✓ 2⇥, consider the following BSDE on the space (⌦ 0 ,F 0 ,P ): Y t = ⇠ T + Z T t [r ✓ s Y s ✓ s Z s ]ds Z T t Z s dB s = ⇠ T + Z T t r ✓ s Y s ds Z T t Z s dB ✓ s ,t2 [0,T], (2.8) where ✓ t = 1 t (r ✓ t r)1, B = B is the canonical Brownian motion under P , and dB ✓ t := ✓ t dt+dB t , t2 [0,T]. We denote the solution by (Y ✓ ,Z ✓ ). Since for each P ✓ 2P K ,wehave P ✓ {| ✓ | K}=1. ThusbyGirsanovtheorem, B ✓ is a P ✓ -Browian motion, and for ✓ t =e R t 0 r ✓ s ds , ✓ t Y ✓ t =Y ✓ 0 + Z t 0 ✓ s Z ✓ s dB ✓ s ,t2 [0,T] is aP ✓ -martingale satisfying ✓ T Y ✓ T = ✓ T ⇠ T .Consequently,wecanwrite P ask,K =sup P ✓ 2 P K E ✓ [ T ⇠ T ]= sup P ✓ 2 P K E ✓ [ T Y ✓ T ]= sup P ✓ 2 P K Y ✓ 0 . (2.9) 85 Letting K!1,andnotingthatP = lim K!1 P K , we have P ask =lim K!1 P ask,K =lim K!1 sup P ✓ 2 P K Y ✓ 0 =sup P ✓ 2 P Y ✓ 0 =sup ✓ 2 ⇥ Y ✓ 0 . (2.10) On the other hand, applying the comparison theorem for BSDEs, it is standard to show that sup ✓ 2 ⇥ Y ✓ 0 =Y 0 ,where Y satisfies the following BSDE on (⌦ 0 ,F 0 ,P ): Y t = ⇠ T + Z T t f(s,Y s ,Z s )ds Z T t Z s dB s , (2.11) where f(t,y,z)=esssup ✓ 2 ⇥ [r ✓ t y ✓ t z]. In other words, we have proved the fol- lowing BSDE representation for the frontier P ask . (Noting that we shall identify P =P 0 .) Theorem 2.5.4. Assume that Assumption 2.2.1 is in force, and assume further that ✓ ⌘ , ✓ 2 ⇥ . Then the best ask price of the LOB can be written as P ask = Y 0 , where Y is the solution to the BSDE (2.11), defined on the canonical space (⌦ 0 ,F 0 ,P 0 ). 2.6 A Principal-Agent Problem View of the Frontier Having studied the frontier of the LOB in details, we now return to the basic assumption of our model, that is, each liquidity provide chooses the position in 86 the LOB so as to compensate the loss of the dividend plan. As a consequence, the frontier of the LOB depends on the particular dividend plan given by the firm, as Theorem 2.4.6 shows. Intherestoftheprojectwillextendtheassumptionintwoways. First,weshall assume that all liquidity providers can also have the option to sell via the market order, and will be accounted as a proportion of the total investment strategy so as to collectively maximize the profit. Second, and most importantly, we shall assume that the collective market order will impact the midprice, and it will in turn a↵ect the firm’s dividend plan. In other words, we will be focusing on the interaction between the sellers (or the liquidity providers) and the firm through the dividend plans. we shall argue that there is a reasonable “cooperative” type of game between the two parties, and can be described as a class of Principal- Agent problem, in which the firm of the underlying asset is the principal and the (collective) LOB liquidity providers is the agent, and the “contract” between them is the dividend plan. The frontier is therefore the agent’s value function based on the “optimal contract”, which in a sense justifies the result in [41]. To formulate the problem, we first recall the dynamics of the “mid-price” (2.1): dX t =b(t,X t ,µ t ,⌫ t )dt+ (t,X t )dB t ,X 0 =x, (2.1) 87 whereµ t is the total number of the market buy orders, and ⌫ t is that of the market sell orders at time t.Wenowmodifythesettingslightlysoastoemphasizethe actions of the liquidity providers. In particular, we shall allow the sellers to use both limit orders and market orders. Assume that the placement of the limit orders is the same as before, but there is an impact of the collective action of the total market orders from all liquidity providers, denoted by u = {u t } t2 [0,T] ,that might a↵ect the midprice, which will be modeled by some standard price impact methodology. We assume that the process u is chosen from the admissible set of strategies denoted asM and takes value in M :=R: M := n u :u is progressively measurable, and u t 0,E h Z T 0 |u s | 2 ds i <1, P 0 a.s. o . Next, let us denote the total liquidity of the (sell) LOB byQ ={Q t } t2 [0,T] ,and consider it as the “nontraded” state variable of the economy (see, e.g., [9,41]), we cansimplyassumethatitisanexogenouscompoundPoissonprocesswithpositive jumps 1 .Anaturalconstraintwouldbethatateachtime t2 [0,T], it holds that 0 Z t 0 u s ds Q t ,t2 [0,T], Pa.s. (2.2) 1 Negative jumps are also possible if we allow, for example, cancellations, although there is no technical di↵erences so long as Q t 0, t2 [0,T] (see, e.g., [51]). 88 We then denote a “retention rate” process through the identity u t Q t :=Q t Z t 0 u s ds, t 0. (2.3) More precisely, measures the unit e↵ect that limit order switched to market order, which could be used as a factor for the adjustment on the dividend plans. Summarizing the discussion above, we now rewrite the dynamics of the return of the mid-price X (2.1) as the following SDE: dX t =[r(t,X,Q t ) p(t,X t ,u t )]dt+ (t,X t )dB 1 t ,X 0 =x, t2 [0,T]. (2.4) Hereintheabove,r(t,q)isthemeanrateofreturn,whichweallowittobea↵ected by the total liquidity; and p(t,x,u)isthe(permanent)priceimpactcausedbythe collective action, u,bythesellers. Weshalldenotethesolutionto(2.4)by X u , when the e↵ect of u needs to be specified, and denote X 0 to be solution to (2.4) when u=0. Remark2.6.1. Weshouldnotethatinthispaperwewillnotspecifythedynamics of the total liquidity Q,butratherkeepitasafactorprocessforthemodeling purpose. Consequently,theconstraints(2.2)andthedynamicsof (2.3)aremerely ad hoc at this point, and they will be simplified when we specify the Principal- Agentproblemlater. Wenotethatingeneral(X,Q)couldbemodeledasaMarkov process, and discussed along the lines of [41]. 89 We shall now follow the ideas of Lehalle and Neuman [47] (see also [16,17,35, 58]) to formulate an optimization problem for the collective seller to maximize the profit by switching a limit order to market order. Such an optimization problem will a↵ect the frontier of the LOB in two ways: (1) through the switching amount u,viatheretentionfactorprocess u ; and(2)throughthepriceimpacttothemid- price X u ,andinturnonthedividendplan. Consequently,weshallarguethatthe value function of such an optimization problem would give the new representation of the frontier of the LOB. To be more precise, we shall now assume that each seller will have three options: (i) holding the stock (not selling at all); (ii) sell with market order; and (iii) sell with limit order. As before, we assume that each seller will choose the placement positiony>x such that she can find an investment strategy H to super-hedge the expected gain of the other two options, less the potential waiting costs or opportunity costs that might incur for staying in the LOB. To quantify the e↵ects of the options (i) and (ii), let us assume that a collective market order process u = {u t } is given. Then, at terminal timeT> 0, the potential gain from the options (i) and (ii) can be calculated respectively as follows: ⇠ (X u T ) u T +C u T := ⇠ (X u T ) u T + Z T 0 [X u t k(t,u t )]u t dt, (2.5) 90 where ⇠ : R + 7! R + is some function representing the dividend plan per share, whichdependsonthemid-priceX u ; u isthepercentageforholdingtheassetafter the market order; and C u T is the for selling the market order, which is calculated using mid-price after the temporary price impact caused by market order u (cf., eg., [16,47]), and k(·,·)issomefunctionrepresentingthepriceimpact. Next, we assume that the cumulative waiting and/or opportunity cost for stay- ing in LOB over [0,T]isofageneralform ⇢ ( u ·^ T ), where ⇢ : C T ([0,1]) 7! R + , satisfying some regularity conditions. Then, for each seller, the placement of the limit ordery>x should be such that it can o↵set the general loss function ⇠ u,x T of the form: ⇠ u,x T := ⇠ (X u,x T ) u T ⇢ ( u ) T +C u T = ⇠ (X u,x T ) u T ⇢ ( u ·^ T )+ Z T 0 [X u,x t k(t,u t )]u t dt. (2.6) where X u,x = X u with X u 0 = x.Inwhatfollowsweshallcallthetriplet(⇠,⇢,k ) defined in (2.6) a generalized dividend plan. Note that if the total market order strategy u is given, then one can follow the same arguments of Theorem 2.4.6 to showthat, foranygivengeneralizeddividendplan(⇠,⇢,k ), thefrontieroftheLOB should be P ask u (x):= inf{y :9H 2H , inf P ✓ 2 P E P ✓ [y+ T ( ¯ V H T ⇠ u,x T )] 0} =sup ✓ 2 ⇥ E P ✓ [ T ⇠ u,x T ] = sup ✓ 2 ⇥ E P ✓ n T ⇠ (X u,x T ) u T T ⇢ ( u ·^ T )+ T C u,x T o , (2.7) 91 where = ✓ as we defined in the previous sections. We note that the expression (2.7) of the “frontier” P ask u (x)amountstosaying that each seller would have to choose the placement of the limit order based on the total market orders from all liquidity providers, which is practically impossible. A more reasonable assumption is that each seller place the limit order in such a way that she can o↵set the loss of a general dividend plan for any possible u2M. That is, one should consider the following set of admissible limit orders: ˜ S(x):={y :9H 2H ,8u2M, inf P ✓ 2 P E P ✓ [x+ T ( ¯ V H T ⇠ u,x T )] 0}. (2.8) Again, we define the frontier of the LOB to be P ask (x) := inf ˜ S=inf{x :9H 2H ,8u2M, inf P ✓ 2 P E P ✓ [x+ T ( ¯ V H T ⇠ u,x T )] 0}. (2.9) Theorem 2.6.2. Assume that Assumptions 2.2.1 and 2.3.1 are in force. Then for a given generalized dividend plan (⇠,⇢,C ), the best asking price (or the frontier of the sell LOB) is given by P ask (x)=sup u2M P ask u (x). (2.10) Proof. The proof is similar to Proposition 2.4.5, we give only a sketch. 92 Throughouttheproofweletxbefixed,andhencedropxfromallthenotations for simplicity. Define p ⇤ := sup u2M P ask u ,weclaimthat p ⇤ 2 ˜ S.Indeed,letting H ⌘ 0, then for any u2M, we have inf P ✓ 2 P E P ✓ [p ⇤ + T ( ¯ V 0 T ⇠ u T )] =p ⇤ +inf P ✓ 2 P E P ✓ [ T ⇠ u f T ]=p ⇤ P ask u 0, (2.11) thanks to (2.7) and the definition of p ⇤.Since u is arbitrary, we conclude that p ⇤ 2 ˜ S,andconsequentlywehave P ask  p ⇤ . Conversely, for any y2 ˜ S,thereexists H ⇤ 2H,suchthat y+inf P ✓ 2 P E P [ T (V H ⇤ T ⇠ u T )] 0, 8u2M. Now, using the “no good deal” assumption on H ⇤ ,wecanthenconcludethat y sup P ✓ 2 P E P ✓ [ T ⇠ u T T ¯ V H T ] sup P ✓ 2 P E P ✓ [ T ⇠ u T ]=P ask u , 8u2M. But this implies that y sup u2M P ask u =p ⇤ .Since y2 ˜ S is arbitrary, this leads to that P ask p ⇤ ,provingthetheorem. Theorem 2.6.2 tells us an important fact. If we denote, for each x2R, J 0 (x;u)=J(0,x;u)=P ask u ,u2M. (2.12) 93 Then the best ask price of the LOB can be written as P ask =sup u2M J(0,x;u)=V(0,x). (2.13) That is, the frontier is the value function of an optimization problem with cost functional J 0 (x;u), with the control being the total market order of the collective sellers. Note that the dividend plan ⇠ is completely determined by the firm, we can then consider this as a corporative gam between the collective seller. In fact, we shall now consider the whole pricing procedure as a Principal-Agent problem, in which the collective seller is the agent, the firm is the principal, and the (gen- eralized) dividend plan is the contract. Consequently, Theorem 2.6.2 then states that the frontier of the sell LOB is actually the value function of the agent (the collective seller) under the optimal “contract”. We now give a more precise definition of the Principal-Agent Problem related to the frontier of dynamical LOB. The Agent Problem. We first recall the retention factor process defined by (2.3). Assuming the total liquidity process Q is semi-martingale, we can write the dynamics (2.3) as d( t Q t )=Q t d t + t dQ t +dh,Q i t =dQ t u t dt, 0 =1. (2.14) 94 But as we pointed out in Remark 2.6.1, in this paper we shall not specify the dynamics of Q,letusassumethatitisapositiveprocesssuchthat Q t >>1atall time. Therefore, from (2.14) we have d t = 1 t Q t dQ t u t Q t dt dh,Q i t , 0 =1. (2.15) SinceQ t >>1and0 1 t  1,weseethat 1 t Qt << 1. Weshallthusapproximate by d t = ˜ u t dt, 0 =1, (2.16) where ˜ u t = ut Qt (which in turn implies thath,Q i⌘ 0). Furthermore, if we assume that the total liquidity satisfies Q t ¯ Q, t2 [0,T], for some constant ¯ Q> 0, then we can consider a compact, convex set U ⇢ R + ,anddefinethesetof admissible controls by: U ad :={u2M :u t 2U; t2 [0,T],a.s.}, (2.17) so that, for any u2U ad ,(2.2)holds,and t =1 R t 0 ˜ u s ds 0, for all t2 [0,T]. Clearly, any U=[0,U 0 ], with 0 > > > > > > > < > > > > > > > > : dX t =[r(t,X t ,Q t ) p(t,X t ,u t )]dt+ (t,X t )dB 1 t ,X 0 =x, t2 [0,T] dC t =[X t k(t,u t )]u t dt, C 0 =0 d t = ˜ u t dt, 0 =1, (2.18) where ˜ u t = u t /Q t , t 2 [0,T], and u 2U ad .InlightofTheorem2.6.2,aswellas (2.12) and (2.13), we define agent’s cost functional to be J A (0,x;u):=sup ✓ 2 ⇥ E P ✓ [⇠ x,u T ],u2U ad , (2.19) and the agent’s value function (i.e., the collective the seller’s minimum selling price) is V A (0,x)= sup u2 U ad J A (0,x;u). (2.20) We note that the agent’s cost functional J A (0,x;u)iscompletelydetermined by ⇠ x,u T defined by (2.6), whence the generalized dividend plan (⇠,⇢,k ). Since the retention cost ⇢ and the temporary price impact k are given by the market, the 96 dividend plan function ⇠ is therefore the deciding factor that is chosen by the firm. We shall now denote J A = J A ⇠ and V A = V A ⇠ when we need to indicate their dependence on function ⇠.Weshallnowfocusonthosefunction ⇠ for which the agent can find the optimal control. That is, for each ⇠ we consider the set M(⇠ ):={u ⇤ 2U ad :J A ⇠ (0.x;u ⇤ )=V A ⇠ (0,x)}, (2.21) and we will be focusing on those ⇠ such thatM(⇠ )6=;.Clearly,foreach ⇠ such thatM(⇠ )6=;,any u ⇤ 2M(⇠ )determinesthefrontier P ask corresponding to the dividend plan xı. The Principal Problem. We now look at the problem from the issuing firm’s perspective. We shall assume that the firm is allowed to choose the dividend plan ⇠ that satisfy the following two natural constraints: (i)M(⇠ ) 6= ;,and(ii) V A ⇠ (0,x)>x+R,whereR>0isaconstant,whichcouldbeinterpretedasagiven spread (on the sell side) that the firm would like to seek in order to push the price higher. In fact, if the frontier gets too close to the mid-price, it might trigger large amount of market orders which will adversely a↵ect the price. More precisely, let us define a set of admissible dividend plans: ⌅:= n ⇠ :M(⇠ )6=;,V A ⇠ (0,x) x+R o . 97 The company will choose an admissible dividend plan so as to maximize an expected utility. In other words, we consider the following Principal problem with cost functional J P (0,x;⇠ ):= sup u2 M(⇠ ) E P [ T U(`(X u,x T ) ⇠ u,x T )], (2.22) and value function V P (0,x):=sup ⇠ 2 ⌅ J P (0,x;⇠ ). (2.23) We note thatM(⇠ ) is the set of all optimal strategies of the agents for the given a dividend plan ⇠ ,anditwillbecomeasingletoniftheoptimalstrategyisunique (in that case the “sup” in (2.22) would disappear). The principal’s goal is then to choose a dividend plan that maximizes the discounted utility of the terminal position,characterizedbythepossiblegainfromtheassetpricinglessthedividend, counting possible price impact from collective sellers optimal strategy. Here we assume that ` is a gain function, and U :R! R is a generic non-decreasing and concave utility function. 98 2.7 A Solved Case. In this section, we will introduce our principal agent pricing view under Cvitani´ c, Possama¨ ı and Touzi’s idea [22]. Moreover, for technical convenience, we may assume that a common agreement on risk-free interest rate process r(t,x,q)is reached. Under fairly general conditions, one can verify that the value function of Agent’s problem is given byV A =v(0, ˜ X 0 ), where the functionv:[0,T]⇥ R 4 ! R canbecharacterizedbytheuniqueviscositysolutionoftheHJB(t,˜ x)2 [0,T]⇥ R 4 where ˜ x := (x,q,,c ) @ t v r t v+H(t,x,q,,c )+ 1 2 2 t @ xx v 2 +L Q v=0, (2.1) where H(t,x,q,,c):= sup u2 M H u (t,x,q,,c)and H u (t,x,q,,c):= n h(t,x,q,u)@ x v+( u)@ v+[x k(t,u)]u@ c v o , with terminal value satisfies v(T,˜ x)= g(˜ x)= c + ⇠ (x) ⇢ ( ) 2 . Usually, it will be very dicult to find a classicalsolutiontothisproblem, whichisarequirementunderCvitani´ c, Possama¨ ı and Touzi’s framework [22]. However, we can consider a di↵usion approximated problem instead of the original problem, which we can obtain a classical solution under certain assumptions. Assumption 2.7.1. (i) permanent trading impact is linear i.e., p(x,u):=pu and temporary trading impact is also linear, i.e., k(t,u):= u , where p 0 and 0 represent the coecients of the impact; 99 (ii) drift term of X is only a↵ected by time t and “nontraded” state variable I, i.e., r(t,x,q):=r(t,q), an example can also be referred to Lehalle and Neuman [47], where they also use a O-U process to denote the imbalance signal of the limit order book; (iii) noise from the signal I is independent with the noise from the underlying asset, i.e., E[dB 1 t dB 2 t ]=0. Theorem2.7.2. Under the assumption (2.7.1), if the final payo↵ ⇠ takes the form (1+d)X T , where d is a given constant called dividend payout ratio relative to the stock price. Then (2.1) becomes @ t v r(t,q)v+r(t,q)@ x v+ 1 2 ( ) 2 @ 2 x v+L Q v 2 +sup u {u(x u )@ c v u@ v pu@ x v}=0, with terminal value V(T,˜ x)= g(˜ x)= c+(1+d)x ⇢ ( ) 2 and there exists a classical solution. Proof. The proof follows the same line in [[17], [47]]. If payo↵ takes the form (1+d)X T ,wecanhavethevaluefunctionas v A := sup u2M E P n T (1+d)X u T u T T ⇢ ( u T ) 2 T Z T 0 ( u t ) 2 dt+ T C u T o , =sup u2M E P n T [(1+d)X u T u T ⇢ ( u T ) 2 Z T 0 ( u t ) 2 dt+C u T ] o , 100 with terminal conditionv A (T,˜ x)=c+(1+d)x ⇢ ( ) 2 . Under the assumption of both temporary and permanent being linear, we can have the corresponding HJB for v A as @ t v A r(t,q)v A +r(t,q)@ x v A + 1 2 ( ) 2 @ 2 x v A +L Q v A 2 +sup u {u(x u )@ c v A u@ v A pu@ x v A }=0. Before we solve the HJB, we need to modify the terminal value and the value function a little bit. Denote ˜ := (1+d) ,˜ ⇢ := ( 1 1+d ) 2 ⇢ and ˜ := ( 1 1+d ) 2 . With some notation abuse, now the value function and terminal value becomes v A := sup u2M E P n T [X u T ˜ u T ˜ ⇢ (˜ u T ) 2 ˜ Z T 0 (˜ u t ) 2 dt+ Z T 0 (X u t u t )u t dt] o , and v A (T,˜ x)= c + x˜ ˜ ⇢ (˜ ) 2 . Now by ansatz we can find a solution v A (t,q,c,x,˜ ;d):= c +x˜ +v(t,˜ ,q ), where we denote v A (t,q,c,x,˜ ;d)asthe value function given a fixed dividend ratio d.Thenifweplugtheansatzsolution into the HJB, we can have @ t v r(t,q)[c+x˜ +v]+r(t,q)˜ +L Q v ˜ ˜ 2 +sup u { u 2 u@ ˜ v pu˜ }=0. 101 Then we can obtain the optimal trading speedu ⇤ = @ ˜ v+p˜ 2 .Pluggingu ⇤ into the HJB, we can have @ t v r(t,q)[c+x˜ +v]+r(t,q)˜ +L Q v ˜ ˜ 2 + 1 2 (@ ˜ v+p˜ ) 2 =0,v(T,˜ ,q )= ˜ ⇢ ˜ 2 . Again by ansatz, we can have v(t,˜ ,q ):= v 0 (t,q)+˜ v 1 (t,q)+˜ 2 v 2 (t,q). Then plug in this solution to the HJB and compare the same order for ˜ ,wecanhave @ t v 0 +L Q v 0 + 1 2 v 2 1 r(t,q)[c+v 0 (t,q)] = 0. for no ˜ terms @ t v 1 +L Q v 1 + 2 v 1 v 2 +[2p r(t,q)]v 1 +[r(t,q) r(t,q)x]=0. for ˜ terms @ t v 2 +L Q v 2 ˜ + 2 v 2 2 +[ 2 p r(t,q)]v 2 + p 2 2 =0. for ˜ 2 terms First we solve v 2 .Ifweobserve v 2 ,wecanhave @ t v 2 +L Q v 2 [ ˜ + 1 8 r(t,q) 2 1 2 r(t,q)p]+ 2 [v 2 + p 1 2 r(t,q) 2 ] 2 =0. Now we define v 2 = p 1 2 r(t,q) 2 +˜ v 2 .Thenwewillhavethefollowing @ t ˜ v 2 +L Q ˜ v 2 [ ˜ + 1 8 r(t,q) 2 1 2 r(t,q)p]+ 2 (˜ v 2 ) 2 =0. ˜ v 2 (T,q)= p 1 2 r(t,q) 2 ˜ ⇢. 102 We can find ˜ v 2 (t,q):=˜ v 2 (t). Then one can have @ t ˜ v 2 + 2 (˜ v 2 ) 2 [ ˜ + 1 8 r(t,q) 2 1 2 r(t,q)p] = 0. Now we can have @ t ˜ v 2 2 [ ˜ + 1 8 r(t,q) 2 1 2 r(t,q)p] ˜ v 2 2 = 2 . Then we can obtain that ˜ v 2 (t)= q 2 [ ˜ + 1 8 r(t,q) 2 1 2 r(t,q)p] 1+⇣e 2 (T t) 1 ⇣e 2 (T t) , = 2[ ˜ + 1 8 r(t,q) 2 1 2 r(t,q)p] and ⇣ = (˜ ⇢ p 1 2 r(t,q) 2 )+ p 2 [ ˜ + 1 8 r(t,q) 2 1 2 r(t,q)p] (˜ ⇢ p 1 2 r(t,q) 2 ) p 2 [ ˜ + 1 8 r(t,q) 2 1 2 r(t,q)p] .Sowecanhave v 2 = p 1 2 r(t,q) 2 + r 2 [ ˜ + 1 8 r(t,q) 2 1 2 r(t,q)p] 1+⇣e 2 (T t) 1 ⇣e 2 (T t) , (2.2) where ˜ := ( 1 1+d ) 2 and ˜ ⇢ := ( 1 1+d ) 2 ⇢ . Next we solve v 1 ,ifwerearrangethe equation we can have @ t v 1 +L Q v 1 [r(t,q) 2p 2 v 2 )]v 1 +[r(t,q) r(t,q)x]=0,v 1 (T,q)=0. Then by Feyman-Kac theorem, we can have v 1 (t,q)=E t n Z T t e R s t [r(u,Qu) 2p 2 v 2 ]du [r(s,Q s ) r(s,Q s )X u ⇤ s ]ds o . (2.3) Similarly, we can also have v 0 (t,i)= E t n R T t e R s t r(u,Qu)du [ 1 2 v 2 1 (s,Q s ) r(s,Q s )C u ⇤ s ]ds o . Therefore, we proved the theorem. 103 Corollary 2.7.3. Given the fixed constant dividend ratio d and assume the exis- tence of fixed risk-free interest rate. The best ask price satisfies the following BSDE Y 0 = Y T + Z T 0 [r(t,Q t )Y t + ( u ⇤ t ) 2 ]dt Z T 0 Z x t (t,X u ⇤ t )dB 1 t Z T 0 Z q t dB 2 t , with u ⇤ t = v 1 (t,q)+2(1+d) u ⇤ t v 2 (t,q)+p(1+d) u ⇤ t 2 (v 1 and v 2 take forms as (2.3), (2.2) respectively) and Y T =C u ⇤ T +(1+d)X u ⇤ T u ⇤ T ⇢ ( u ⇤ ) 2 . Since we showed the existence of the classical solution to the agent problem, under the idea in [22], we obtain by Itˆ o’s formula the following representation of g(˜ x):=c+(1+d)x +⇢ 2 : g( ˜ X T )= v(0, ˜ X 0 ) Z T 0 [r(t,Q t )V t + ( u ⇤ t ) 2 ]dt+ Z T 0 Z x t (t,X t )dB 1 t + Z T 0 Z q t dB 2 t , (2.4) where z t := Dv(t,˜ x t )and u ⇤ is the optimal control to the corresponding control problem. Definition 2.7.4. We denote V as the collection of all F predictable process Z : [0,T]⇥ ⌦ ! R 2 satisfying 104 • ||(Y,Z)|| 2 <1, where for an initial value Y 0 2R, the process Y Z is defined P 0 a.s. by the (2.4), where ||(Y,Z)|| 2 :=E[|Y ⇤ T | 2 + R T 0 |Z x t | 2 dt+ R T 0 |Z q t | 2 dt] . • There exists a u Z 2M such that the following equality holds dt⌦ P 0 a.e. on [0,T]⇥ ⌦ , H(t,X t ,Q t , t ,C t )= n h(t,X t ,Q t ,u z t )@ x Y Z t +( u z t )@ Y Z t +[X t k(t,u z )]u z t @ c Y Z t o . Nowwearereadytointroduceourprincipalproblemundertheframework[22]. We introduce V(Y 0 )=sup Z2V sup u ⇤ 2 M ⇤ E P [ T U(l(X T ) Y Z T ⇢ ( u ⇤ T ) 2 C u ⇤ T )] and Y Z T is constructed in the sense of (2.4). Furthermore, we specifically consider the case ⌅ := {(1+d)X T : M ⇤ ((1+d)X T ) 6= ;, u ⇤ T 6=0,8u ⇤ 2M ⇤ ,V A x 0 }.The assumption u ⇤ T 6=0,8u ⇤ 2M ⇤ means that we only consider scenarios that the spread still exist. Proposition 2.7.5. Assume thatV6= ;, Then⌅ = { Y Z T C u ⇤ T ⇢ ( u ⇤ T ) u ⇤ T : Y 0 R,and,Z2V}. In particular, V P =sup Y 0 R V(Y 0 ). Proof The proof follows the same line in [22]. We shall prove the theorem in following steps. 105 Step 1. For any (1+d)X T 2⌅, we want to show that the following BSDE has an unique solution Y 0 =Y T + Z T 0 [r(t,Q t )Y t + ( u ⇤ t ) 2 ]dt Z T 0 Z x t (t,X t )dB 1 t Z T 0 Z q t dB 2 t , (2.5) where Y T := C u ⇤ T +(1+d)X u ⇤ T u ⇤ T ⇢ ( u ⇤ T ) 2 ,and u ⇤ can be derived in proof of Theorem 2.7.2 . With the assumption that E P 0 [ R t 0 |u ⇤ s | 2 ds] < 1 and the condi- tion X 2 L 2 F ([0,T]⇥ ⌦), we can conclude that (2.5) has an unique solution and E P 0 [ R T 0 |Z x t | 2 dt+ R T 0 |Z q t | 2 dt]<1. step2. We want to show that Y 0 = V A (Y Z T ). From step 1, we know that the terminal value C u ⇤ T +(1+d)X u ⇤ T u ⇤ T ⇢ ( u ⇤ T ) 2 can be treated as a terminal value Y Z T corresponding to a BSDE. Moreover by Itˆ o’s formula, one can derive E P 0 [ T Y Z T T Z T 0 ( u ⇤ t ) 2 dt]= Y 0 E P 0 h T Z T 0 n ˜ H t dt Z t d ˜ X t 1 2 t dh ˜ Xi t +(r(t,Q t )Y t + ( u ⇤ t ) 2 )dt oi , where we denote ˜ H t := ˜ H(t,˜ x,v(t,˜ x),Dv(t,˜ x),D 2 v(t,˜ x)) = H(t,x,q,,c ) r(t,q)v + 1 2 2 t D 2 x v ( 2 )+L Q v.Inspiringfromtheaboveequation,forany admissible trading strategy u,wecanrewritethecostfunctionoftheagentas J A (u,Y Z T )=Y 0 E P 0 h T Z T 0 [H(t,X t ,Q t , t ,C t ) H u (t,X t ,Q t , t ,C t )]dt i . 106 By the definition of the Hamiltonian H in (2.1), we can claim thatJ A (u,Y z T ) Y 0 , and thusV A (Y Z T ) Y 0 for the arbitrariness ofu2M.Atthesametime,weknow that u ⇤ can be obtained such that J A (u ⇤ ,Y Z T )=Y 0 .Therefore,wecanverifythat Y 0 =V A (Y Z T )andcompleteourproof. We can see that the principal problem isV(Y 0 )isastandardstochasticcontrol probleem under control processesZ and controlled state processes ( ˜ X,Y Z ). In the Markovian case where the dependence is only through the current value, we can see that the relevant optimization term for the dynamic programming equation corresponding to the control problem V is defined for (t,˜ x,y) 2 [0,T]⇥ R 4 ⇥ R where ˜ x := (x,q,,c ) G(t,˜ x,y,N,M):= sup z2 R 2 sup u ⇤ 2M ⇤ n [r(t,q) pu]N x +f(t,q)N q uN +(x u )uN c [r(t,q)y+ 2 ]N y + 1 2 2 (t,x)M xx + 1 2 ( q ) 2 M qq +[ 1 2 2 (z q ) 2 M yy + 1 2 (t,x) 2 (z x ) 2 M yy ]+ (t,x) 2 z x M xy + 2 z q M qy o . If we assume the existence of the maximizer z ⇤ (t,˜ x,y,N,M) of the Hamiltonian G(t,˜ x,y,N,M). Then we will have the standard verification result in the Marko- viancase. WeshalldenotebyT T thecollectionofallF stoppingtimeswithvalues in [0,T]. 107 Proposition 2.7.6. In a Markovian setting, let v2C 1,2 ([0,T]⇥ R 5 ) be a classical solution of the dynamic programming equation 8 > > > < > > > : (@ t v r t v)(t,˜ x,y)+G(t,˜ x,y,Dv(t,˜ x,y),D 2 v(t,˜ x,y)) = 0,(t,˜ x,y)2 [0,T)⇥ R 4 ⇥ R v(T,˜ x,y)=U(l(x) y ⇢ 2 c), ˜ x := (x,q,,c ). Assume further that • The family {v(⌧, ˜ X ⌧ ,Y ⌧ ))} ⌧ 2T T isP-uniformly integrable for all u2M ⇤ (Y Z T ) and all Z2V. • The function G has a maximizer z ⇤ such that Z ⇤ 2V. Then, V(Y 0 )=v(0, ˜ X 0 ,Y 0 ), and Z ⇤ is an optimal control for the problem V. Proof Details can be referred to [22]. 108 Chapter 3 Deep Signature FBSDE Algorithm 3.1 Introduction Motivation. Recent developments of numerical algorithm for solving high dimensional PDEs draw a great amount of attention in various scientific fields. In the seminal paper [76], deep learning technique was first introduced to study the numerical algorithms for high dimensional parabolic PDEs. The deep learning BSDEmethodisbasedonthenon-linearFeynman-Kacformula,whichprovidesthe equivalent relations between parabolic PDEs and Markovian backward stochastic di↵erential equations (BSDEs) (see e.g. [60]). When the non-Markovian property, e.g. path-dependent property, is involved, the BSDE is equivalent to a path- dependent PDE (PPDE), which was first introduced in [26] for path-dependent option pricing problem. The deep learning BSDE method has been recently extended to design numerical algorithms for PPDEs. The non-Markovian prop- erty introduces extra complexity in the numerical scheme, and it returns a high 109 dimensional problem even if the original space variable is low dimensional. In this study, we shall focus on the numerical solutions for the corresponding Markovian and non-Markovian FBSDEs. For the deep learning BSDE method [76], it shows the eciency of machine learninginsolvinghighdimensionalparabolicPDEsbutsubjecttosmallLipschitz constants or equivalently small time duration. The exponential stopping time strategy has been introduced in [72] to extend the time duration. However, both algorithms are still using the deep neural network combined with standard Euler scheme in essence, which makes it sensitive to the time discretization. Namely, the time dimension is still large for long time duration, which may take a long time to trainthedeepneuralnetwork(DNN)model. Furthermore,thedeeplearningBSDE methodisnotrobusttomissingdata. Ifwemissaproportionofourdata(e.g. data points in the Euler scheme), the accuracy will be a↵ected. In particular, this is the sametypeofdicultywhendealingwithhighfrequencydata. Inthiscase,onehas todown-samplethestreamdatatoacoarsertimegridtofeeditintotheDNN-type algorithm. It may miss the microscopic characteristic of the streamed data and render lower accuracy. On the other hand, the high frequency and path-dependent features show up naturally in option pricing problems and non-linear expectations within various financial contexts, e.g. limit order book [11,15,20,38,52], nonlinear pricing [60,77], Asian option pricing [57], model ambiguity[8,10,21,34], stochastic games and mean field games [32,66], etc. 110 Ourwork. Motivatedbytheseproblems, weintroducethedeepsignaturetrans- formation into the recurrent neural network (RNN) model to solve BSDEs. The “signature” is defined as an iterated integral of a continuous path with bounded p-variation, forp> 1, which is a recurring theme in the rough path theory intro- duced by T. Lyons [50]. The “signature” has recently been used to define kernels [19,45,55] for sequentially ordered data in the corresponding reproducing ker- nel Hilbert space (RKHS). This idea is further developed in [44] to design “deep signature” by combing the kernel method and DNN. Furthermore, the “deep sig- nature” has been used in RNN to study controlled di↵erential equations in [49]. The signature approach also provides a non-parametric way for extraction of char- acteristic features from the data, see e.g. [48]. The data are converted into a multi-dimensional path through various embedding algorithms and then processed for computation of individual terms of the signature, which captures certain infor- mation contained in the data. The advantage of this signature method is that this method can deal with high frequency data, and is not sensitive to the time discretization. Motivated by this idea, we propose to combine the signature/log- signaturetransformationandRNNmodeltosolvetheFBSDEs, whichshouldhave amuchcoarsertimepartition,abetterdownsamplinge↵ect,andmorerobustto the high-frequency data assumptions. 111 Related works. The numerical algorithm for solving PPDE with path- dependent terminal condition (first type PPDE) has been recently studied in [73,74] by using recurrent neural network. The second type PPDE arises from the Volterra SDE setting, where the non-Markovian property is introduced by the forward process instead of the terminal condition. The numerical algorithms for the option pricing problem in the Volterra SDEs setting has been recently studied in [40,72] by using deep learning, [6] by using regularity structures, and [28] by using cubature formula. However, none of these works consider the high frequency data features in the algorithm. Neither do they consider the longer time duration in the model. Furthermore, wealsoprovidetheconvergenceanalysisofouralgorithmafterintro- ducing the signature/log-signature transformation layer into the RNN model. 3.2 Algorithms and convergence analysis We consider the Markovian and non-Markovian FBSDEs of the following form, for t2 [0,T], (M) 8 > > > < > > > : X t =x+ R t 0 b(s,X s )ds+ R t 0 (s,X s )dW s , Y t =g(X T )+ R T t f(s,X s ,Y s ,Z s )ds R T t Z s dW s , (3.1) 112 and (NM) 8 > > > < > > > : X t =x+ R t 0 b(s,X s )ds+ R t 0 (s,X s )dW s , Y t =g(X · )+ R T t f(s,X · ,Y s ,Z s )ds R T t Z s dW s . (3.2) In both the Markovian (M)andnon-Markovian(NM) FBSDEs system above, we denote {W s } 0 s T as R d -valued Brownian motion. Throughout the paper, unless otherwise stated, the process X,Y,and Z take values in R d 1 ,R d 2 , and R d 2 ⇥ d ,respectively. Wedenote g(X T )asthestatedependentterminalcondition and denote g(X · ) as the terminal condition depending on the path of X, which corresponds to the the payo↵ function in the option pricing problem. The pair (Y t ,Z t ) 0<t<T solves the BSDE in (M)and(NM) respectively. We present signature/ log-signature FBSDE numerical scheme in detail. We firstpartitionthetimehorizon[0,T]intontimestepswithameshsize t :=T/n, and the time partition is given by 0 = t 0  t 1 <···<t n = T.Thestateprocess X is generated from Euler scheme as X n t i+1 ⇡ X n t i +b(t i ,X n t i ) t+ (t i ,X n t i ) W t i+1 , (3.3) 113 where W t i+1 := W t i+1 W t i denotes the increment of the Brownian motion. Next, for some k2{a 2 Z + : n/a 2 Z + },wepartitionthetimeinterval[0,T] into ˜ n := n/k segmentations with step size u := k t.Thesegmentationcan be written as 0 = u 0 "do 4: Randomly select a mini-batch of data, with batch size M. 5: for i2{1,··· ,˜ n} 6: Z j,✓, Sig u i =R ✓ (⇡ m (Sig(X j,n ) 1 ),···,⇡ m (Sig(X j,n ) i )). 7: Compute Y j,˜ n,Sig u i+1 from Euler scheme (3.4). 8: end 9: Compute loss(✓,Y 0 ,Z 0 )= 1 M P M j=1 (Y j,˜ n,Sig T g(X j,n T )) 2 for problem (M). loss(✓,Y 0 ,Z 0 )= 1 M P M j=1 (Y j,˜ n,Sig T g(X j,n [0,T] )) 2 for problem (NM). 10: Minimize loss, and update ✓ by stochastic gradient descent. Assumption 3.2.1. Let the following assumptions be in force. • b,,f,g are deterministic taking values inR d1 , R d1⇥ d , R d2 , R d2 , respectively; and b(·,0), (·,0),f(·,0,0,0) and g(0) are bounded. • b,,f,g are smooth enough with respect to all variables (t,x,y,z) and all derivatives are bounded by constant L. We are now ready to present the universality approximation property of deep signature/log-signature Markovian BSDE. 115 Lemma 3.2.2. Let Assumption 3.2.1 hold and assume kh is small enough, for some constantC> 0, and for any">0, there exists recurrent neural networkR ✓ , such that ˜ n 1 X i=0 E h Z u i+1 u i |Z t Z ✓, Sig u i | 2 dt i  C[1+|x| 2 ]kh+2T". Furthermore, we have the following estimate. Theorem 3.2.3. Let Assumption 3.2.1 be in force. Then one can conclude that, max 0 i ˜ n E[sup u i  t u i+1 |Y t Y ˜ n,Sig u i | 2 ] C[1+|x| 2 ]kh. The same estimates follow after replacingZ ✓, Sig · withZ ✓, LS · in Lemma 3.2.2 and Theorem 3.2.3. 3.3 Results In this section, we implement our algorithm to a wide range of applications includ- ing European call option, lookback option under Black-Scholes model, European call option under Heston model, a high dimensional example, and a nonlinear 116 example. In summary, our Sig/LS-FBSDE method has the following advantages over other numerical methods in the current literature 3 : 1. Our algorithm is capable to find a more accurate solution to the FBSDE. 2. Ouralgorithmiscapabletoapproximatethetruesolutionecientlyinterms of time. 3. Our algorithm is capable to handle high frequency data in a long time dura- tion. The results are accurate and computation times are ecient. 4. Ouralgorithmiscapabletohandlehighdimensionalandnon-linearscenarios. 3.3.1 Best Ask Price for GBM European Call Option We first apply our algorithm to provide lower/upper bound for the limit order book spread. Let the underlying asset X follows a geometric Brownian motion, dX t =X t (r t dt+ t dW t ),X 0 =x 0 , 3 Thedesktopweusedinthisstudyisequippedwithani7-8700CPUandaRTX2080TiGPU. For all the examples in this paper, we generated in total of ˆ N = 100,000 paths for the forward processes. 1,000 paths were used to test, and the rest were used to train the neural network. 117 where W t is a standard Brownian motion under risk neutral measureQ.Basedon the no good deal theory [11] [Theorem 3.9] , the best ask price for the European call option at level  can be represented as P ask, =sup P2Q ngd, E P [ T 0 (X T K) + ], (3.5) where Q ngd, denotes no good deal pricing set at level  ,and T 0 is a discount factor. With the specification of the pricing measureQ ngd, 4 , under no good deal conditionassumption(see[11]Definition2.16andTheorem2.17), thebestask/bid price (3.5) at level  are unique solutions to the following BSDEs at t=0, Y ± t = ⇠ T + Z T t G ± (X t ,Y t ,Z t )dt Z T t Z t dW t , (3.6) where ⇠ T := (X T K) + and G ± (x,y,z)= ± ||z|| ry.Inthisexample,we implement 1-dimensional best ask scenario for (3.6), and we compare results from our signature methods with simple neural network method. We choose the follow- ing parameters for the simulation x 0 =100.0, =0.20,r=0.05, =0.05,K = 80,T =1,m=3,˜ n=5,andbatchsize1000. Table 3.1: Best ask price for GBM European call option 4 See details in the Appendix. 118 Simple NN Sig-LSTM Sig-LSTM Sig-LSTM Sig-LSTM n=100 ˜ n=5, n=100 ˜ n=5, n=500 ˜ n=5, n=1000 ˜ n=5, n=5000 25.526 25.48 25.46 25.46 25.45 AsweseefromTable3.1, ouralgorithmcombiningsignaturewithLSTMneural network (labeled as Sig-LSTM) outperforms the simple neural network method in terms of eciency, our algorithm runs 20 times faster than simple neural network approach with n=100. Thisiswhatweshouldexpect,sinceforeachiteration our algorithm runs 5 steps segmented by signature (˜ n=5)insteadof100steps (n = 100) in the simple neural network approach with Euler scheme. Also, as we can see in Table 3.1, the result converges to 25.45 when n increases. More accuracy and time eciency results comparisons are illustrated in the lookback option example, which is a path dependent option. 3.3.2 Lookback Option Example In this example, we consider the lookback option pricing under the the classical Black-Scholes model with constant interest rater,andvolatility . Under the risk 119 neutralmeasureQ,thestockprices(X t ) t 0 andthelookbackoptionpriceY t follow the FBSDEs below, 8 > > > < > > > : dX t =rX t dt+X t dW t ,X 0 =x 0 , dY t =rY t dt+Z t dW t ,Y T =X T inf 0 t T X t . In this example, we choose the following parameters in simulation, x 0 =1, = 1,r=0.01,T =1,m = 3. In Figure 3.1, we compare the convergence of look- back option prices from di↵erent methods, and di↵erent time discretization steps. Vanilla-LSTM refers to the algorithm that the inputs to the neural networks are the stock prices. PDGM from [74] is a numerical scheme based on recurrent neural network, and it is used to solve PPDEs. LogSig-LSTM and Sig-LSTM refer to the two numerical algorithms proposed in this study. Figure 3.2 and Figure 3.3 list all computation errors and running times over di↵erent methods and time steps respectively. The first observation is that under the same number of time steps, the numer- ical solutions from all methods are very similar. Secondly, the key to improve the numerical solutions to be closer to the true solution is the number of the time steps during simulation, which is quite intuitive. The smaller the mesh size during simulation, the result is better. With n=5000,ourlog-signatureandsignature 120 perform the best, and with ˜ n=20,thenumericalsolutionisonlyapproximately 0.6% apart from the true solution. The third observation is that the convergence rate is slower with smaller number of segmentations ˜ n in log-signature and signa- ture method. In addition, with a larger number of segmentations, the numerical results are generally better. Therefore, one may be encouraged to have n become as large as possible. However, this is not feasible in practice due to the running times. Figure 3.3 compares the running times over di↵erent methods and time steps respectively. The running times are approximately linear with the number of seg- mentations and time steps. Log-signature and signature methods run 100 times faster with 5 segmentation (˜ n =5)thanvanilla-LSTMwith500timessteps (n=500). Therefore,summarizingthestockdatapathsintosignatureintoa few segmentations, and then inputting them into the neural network would save us a great amount of time, and obtain the similar accuracy. In addition, our method can handle high-frequencey data. It would be imprac- ticable to input a stock paths with n = 5000 into the vanilla-LSTM since it would take too long to train. However, we could first divide the 5000 time steps into 5 or 20 segmentation, and then compute the log-signature and signature of segmen- tations, which will be finally input into the neural networks. As we can see from this example, our method reaches a higher accuracy in an time ecient manner. 121 Figure 3.1: Convergence on lookback option prices (T =1)viadi↵erent methods. Figure 3.2: Option pricing errors across di↵erent methods and time steps. Figure3.3: Computationtimesoverdif- ferent methods and time steps. Figure 3.4: High frequency long dura- tion lookback option pricing example (T =10). Figure 3.5: High frequency long dura- tion lookback option pricing example (T =10). Figure 3.6: A better bound for bid and ask prices. 122 Furthermore, our method could handle high frequency data with a long time duration. Continuing with lookback option example, now we choose the parameters to be x 0 =1,r =0.01, =0.05,T =10. Sincethenumerical di↵erence between log-signature and signature methods are minimal, we only make a comparison between vanilla-LSTM and Sig-LSTM in Figure 3.4. Figure 3.5 plots a closeup of lookback option prices with di↵erent time-steps. Comparing to Vanilla-LSTM with n=500,ourSig-LSTMmethodswith˜ n=5and n=5000 improves the accuracy by 1.36%, and underestimates the solution only 0.624%. In the meantime, our Sig-LSTM method with ˜ n=5and n=5000runs100times faster than Vanilla-LSTM with n=500,whichisquiteimpressive. 3.3.3 European Call Option in the Heston Model under Parameter Uncertainty In this section, we study the European call option pricing problem with stochastic volatility model under parameter uncertainty. The general model setup follows 123 from [21], where the asset priceS and the variance process V in the Heston model are given by 8 > > > < > > > : dS t =rS t dt+ p V t S t (⇢dW 1 t + p 1 ⇢ 2 dW 2 t ), dV t =(✓ [ + ]V t )dt+ p V t dW 1 t , where we denote W 1 ,W 2 as two independent Brownian motions under risk neu- tral measure Q,withcorrelation ⇢ 2 ( 1,1). Under the parameter uncertainty situation, we do not know the precise values of (r,, ), with = ✓ . However, following the ellipsoidal specification 5 of uncertainty, the pricing bound for the Heston call option under model ambiguity is the unique solution of the following BSDE with payo↵ g(S T )=(S T K) + at maturity T, 8 > > > < > > > : dY ± t = H ± (S t ,V t ,Y ± t ,Z t )dt+Z t dW t ,Y ± T =g(S T ), H ± (S t ,V t ,Y ± t ,Z t )=± p ⌘ T t ⌃ T ⌘ t rY t , (3.7) where ⌘ t 2 R 3⇥ 1 can be computed explicitly. We implement the example in [21] withthesameexperimentsetup: S 0 =100,V 0 =0.0457,r=0.05, =5.070,✓ = 5 See details in the Appendix. 124 0.0457, =0.4800, ⇢ = 0.767,K =100,T =1,andcovariancematrix ⌃ = Diag(2 .5e 05,0.25,1e 04)2R 3⇥ 3 . From Figure 3.6, we can see that our method provides a better pricing bound over the recursive MARS degree 2 with variance reduction method (denoted as ”MARS”inthefigure)in[21],byprovidingaslightlywiderboundfortheoptimally controlled value process. The closeup plots are in Figure 3.7. As we increase the number of time steps to n = 200, the Vanilla-LSTM performs better than MARS method with n=25,and n=100. Lastly,with˜ n=5, and n=5000,our Sig-LSTM method eciently improves the bound. This is what we should expect. With a larger number of time discretizationn,thedriverH ± in (3.16) are updated moreaccurately,whichleadstothevalueprocessY ± in(3.16)optimisedtoahigher degree. Lastly, we should mention that traditional numerical methods like ”MC” and”MARS”arenotecienttobeextendedtolargetimediscretizationscenarios. Table 3.2: Bid ask prices for European call under Heston model MARS Bid MARS Bid Vanilla-LSTM Bid Vanilla-LSTM Bid Sig-LSTM Bid Sig-LSTM Bid Sig-LSTM Bid n=25 n=100 n=100 n=200 ˜ n=5, n=100 ˜ n=5, n=500 ˜ n=5, n=5000 9.74 9.62 9.59 9.58 9.53 9.527 9.50 MARS Ask MARS Ask Vanilla-LSTM Ask Vanilla-LSTM Ask Sig-LSTM Ask Sig-LSTM Ask Sig-LSTM Ask n=25 n=100 n=100 n=200 ˜ n=5, n=100 ˜ n=5, n=500 ˜ n=5, n=5000 12.16 12.25 12.17 12.41 12.48 12.52 12.57 125 Figure 3.7: Closeup plots for bid and ask prices. 3.3.4 High Dimensional and Non-linear Example In this section, we consider the following path-dependent BSDE, Y t =g(X · )+ Z T t f(s,X · ,Y s ,Z s )ds Z T t Z s dB s , (3.8) and the forward process is given by dX t = dB t ,X 0 =0. Basedontheassoci- ation with PPDE and the nonlinear Feynman-Kac formula, we construct a high dimensionalexampleandanon-linearexamplebelow, whichwecouldfindthetrue solution. High Dimensional Example For simplicity, let f =0, and g(X · )= ⇣ R T 0 P d i=1 X i s ds ⌘ 2 , we can find the explicit solution of this problem. We then compare the true solution with the solution approximated by our algorithm. In this example, we use the deep log-signature BSDE algorithm because the input of the network grows exponentially in terms of the dimension. With 126 d=20,T =1,˜ n=5,n=100,after10000trainingiteration,theapproximated solution of Y 0 from our algorithm is 6.60 with an error of 1% to the true solution of 6.66. Again, our algorithm runs only 5 steps (˜ n = 5) during training, which is quite time ecient. Here we remark that even our algorithm is able to approx- imate the true solution of the high dimensional example, it is more suitable for highfrequency, pathdependentandlongdurationdata. Thisisbecausewhengen- erating signatures / log-signatures from high dimensional paths, the dimension of signatures / log-signatures would increase exponentially in terms of the dimension of the path, we could see details in the Appendix. Nonlinear Example In this example, we apply our algorithm to approximate the solutions of an non-linear FBSDE (3.8) with d=1,andthegenerator f(t,X [0,t] ,Y t ,Z t )= ✓ min µ2 [µ,µ] µZ t +max(sin(X t + Z t 0 X s ds),0)(µ+X t ) (3.9) +min(sin(X t + Z t 0 X s ds),0)(µ+X t )+ 1 2 cos(X t + Z t 0 X s ds) ◆ In the numerical implementation, we choose the terminal condition to be Y T = cos(X T + R T 0 X s ds). This example is inspired by a two person zero sum game from [66]. We choose the following parameters to implement our algorithm: X 0 =0, µ=0.2, µ=0.3, T = 1. As illustrated in Figure 3.8 and Table 3.3, with an increase of number of segmentation ˜ n and number of time steps n in the Euler 127 scheme will simultaneously improve the accuracy. With only 20 segmentations (˜ n=20)for n=1000, Y 0 reaches 0.9982 with an error of only 0.18%, where true solution is 1. Figure 3.8: Nonlinear Example. Table 3.3: Y 0 in the Nonlinear Example ˜ n=5 ˜ n=20 ˜ n=50 ˜ n=100 n=100 0.986 0.9979 0.9988 – n=1000 0.987 0.9982 0.9991 0.9997 3.4 Conclusion This paper aims to develop ecient algorithms to solve non-Markovian FBSDEs or equivalent PPDEs. We combine the signature/log-signature transformation together with RNN model to solve the FBSDE numerically. Our algorithms show advantages in solving path-dependent problems, high-frequency data problems, 128 and long time duration problems, which apply to a wide range of applications in financial markets. 129 3.5 Appendix 3.5.1 Signature and signature transformation In this section, we introduce the preliminary facts about the signature from the rough path theory [50] and the signature transformation [49] we used in the algo- rithm. Ingeneral,foraboundedvariationpathx t 2R d ,fort2 [0,T],thesignature of x (up to order N) is defined as the iterated integrals of x.Moreprecisely,fora word J=(j 1 ,···,j k )2{1,···,d} k with size|J| =k, Sig N (x) t = N X k=0 Z 0<t 1 <···<t k <t dx t 1 ⌦···⌦ dx t k ,t2 [0,T], (3.10) = ⇣ 1, d X j=1 Z t 0 dx j t 1 ,··· , X |J|=N Z 0<t 1 <···<t N <t dx j 1 t 1 ···dx j N t N ⌘ where we use the convention that Sig 0 (x) t ⌘ 1. The signature Sig N (x) t lives in astrictsubspace G N (R d ) ⇢ T N (R d ), known as the free Carnot group over R d of step N,where T N (R d )= N k=0 (R d ) ⌦ k is the truncated tensor algebra over R d . Furthermore, theexponentialmapdefinesthedi↵eomorphismfromtheLiealgebra g N (R d )totheLiegroupG N (R d ), namely G N (R d )=exp(g N (R d )), (3.11) 130 where g N (R d )istheLiesub-algebraof T N (R d )generatedbythecanonicalbasis e i ,i=1,...,d, ofR d ,andtheLiebracketisgivenby[a,b]=a⌦ b b⌦ a.Thus, the log signature lives in the linear space g N (R d ), and we we denote logarithm of the signature of the path x as LS(x). Let ⇡ m (·)betheprojectionmapofthe signature and the log signature at order m.Wedenote LS m (X)= ⇡ m (log(Sig(x))) as the truncated log signature of a pah x of order m. We introduce the following standardtreatmentwhencomputingthesignatureofapathtogetherwiththetime parameter. Definition 3.5.1. Given a path x:[a,b]! R d , we define the corresponding time- augmented path by ˆ x t =(t,x t ), which is a path in R d+1 . We should remark here that a bounded p-variation path is essentially deter- mined by its truncated signature at order bpc.Thismeansthatessentiallyno information is lost when applying the signature transform of a path at certain order without using the whole signature process. Proposition 3.5.2 (Universal nonlinearity, [2], see also [44] Proposition A.6). Let F be a real-valued continuous function on continuous piecewise smooth paths inR d and let K be a compact set of such paths. Then for all x2K and 8✏, there exists a linear functional L such that for all x2K, |F(x)L (S(ˆ x))| ✏. (3.12) 131 We introduce the signature and the log signature layer in [49]. Definition 3.5.3 (Signature and log Signature Sequence Layer). Consider a dis- crete d-dimensional time series (x t i ) n i=1 over time interval [0,T].A (log) signature layer of degree m is a mapping fromR d⇥ n toR ˆ d⇥ N , which computes (l k ) N 1 k=0 as an output for any x, where l k is the truncated (log) signature of x over time interval [u k ,u k+1 ] of degree m as follows: l k = LS m [u k ,u k+1 ], (or S m [u k ,u k+1 ]) where k2{0,1,...,N 1} and ˆ d is the dimension of the truncated (log) signature. 3.5.2 Backgrounds of numerical examples Throughout this section, we denote (⌦ ,F,(F t ) t 0 ,P)asthefilteredprobability space and denoteQ as the risk neutral measure. Best Ask Price for GBM European Call Option. The limit order book spread has been extensively investigated through No Arbitrage Bound/No Good Deal bound in incomplete markets [15,18,20,38,52,54]. Traditionally, under a risk neutral measure Q, one may assume the underlying asset X follows a geometric Brownian motion i.e., dX t =X t (r t dt+ t dW t ),X 0 =x 0 , 132 where W t is a standard Brownian motion under Q.Bynogooddealtheory,the best ask price for the European call option at level  ( can be thought as the bound for girsanov kernels) can be represented as P ask, =sup P2Q ngd, E P [ T 0 (X T K) + ], (3.13) where the set Q ngd, is nonempty and called the no good deal pricing set at level  , T t is the discount factor defined by fixed risk-free interest rater t .Moredetails can be referred to [11,18,52]. In our setting, we defineQ ngd, as follow Q ngd, := n Q ✓ : dQ ✓ dQ =M T ( ✓ ,W); sup t2 [0,T] || ✓ t ||  o , (3.14) where M T ( ✓ ,W):= exp{ R T 0 ✓ t dW t 1 2 R T 0 || ✓ t || 2 dt},theprocess ✓ t denote all possible girsanov kernels and their bound is  . With the specification of the pricing measure setQ ngd, in (3.14), we can show that (3.5) is closely linked to the following BSDE. The proof follows from the comparison theorem for BSDEs, and we refer details in [62]. Theorem 3.5.4. Assume no good deal assumption, then one can obtain that the best ask/bid price (3.5) at level  are unique solutions to the following BSDEs when t=0 Y ± t = ⇠ T + Z T t G ± (X t ,Y t ,Z t )dt Z T t Z t dW t , (3.15) 133 where ⇠ T := (X T K) + and G ± are optimised drivers over all possible kernels in (3.14), G (x,y,z)= min || ✓ ||  f(x,y,z, ✓ ) and G + (x,y,z)= max || ✓ ||  f(x,y,z, ✓ ). Here we define f(x,y,z, ✓ ):= ry +z ✓ . Furthermore, we can obtain the opti- mised drivers as follow G ± (x,y,z)=± ||z|| ry. Lookback options. In this example, we consider the classical Black-Scholes model setting. Under the risk neutral measureQ,thestockprices(X t ) t 0 follows ageometricBrownianMotionwithconstantinterestrate r,andvolatility , dX t =rX t dt+X t dW t . Lookback option is one of the path-dependent financial derivatives. A lookback call option with floating strike is given by the payo↵ function g(X [0,T] )=X T inf 0 t T X t . 134 It is clear that the option price Y t has the form Y t =e r(T t) E Q [g(X [0,T] )|F t ]. Fortunately, Y t has an explicit solution, (e.g. [57]), Y t =X t ( a 1 ) m t e r(T t) ( a 2 ) X t 2 2r ( a 1 ) e r(T t) ✓ m t y t ◆ 2r/ 2 ( a 3 ) ! , where m t := inf 0 u t X t ,and a 1 = log(X t /m t )+(r+ 2 /2)(T t) p T t ,a 2 =a 1 p T t and a 3 =a 1 2r p T t. In the meantime, the option price Y t can also be represented as a solution to the following BSDE, 8 > > > < > > > : dY t = rY t dt+Z t dW t , Y T = X T inf 0 t T X t . Therefore, we are able apply our numerical method, and compare solutions with the true solution, and solutions from other numerical schemes. EuropeancalloptionsintheHestonmodelunderparameteruncertainty. Recall that, under the risk neutral measure Q, we consider the following Heston 135 model in [21] for a European call option pricing problem with stochastic volatility model under parameter uncertainty. For t2 [0,T], the asset price S and forward variance process V follows, 8 > > > < > > > : dS t =rS t dt+ p V t S t (⇢dW 1 t + p 1 ⇢ 2 dW 2 t ), dV t =(✓ [ + ]V t )dt+ p V t dW 1 t , and W 1 ,W 2 are two Brownian motions under Q with correlation ⇢ 2 ( 1,1). Parameters (,✓, ) are assumed to be nonnegative and satisfy the Feller’s condi- tion 2✓ 2 to guarantee that variance process V is bounded below from zero. Moreover, under the parameter uncertainty situation, an elliptical uncertainty set for parameters (r,, ,where ⌘ ✓ )with(1 ↵ ) confidence is given by the quadratic form U = {u : u T ⌃ 1 r,, u  },where u is the perspective deviance towards the true parameters denoted as u t =(r t r, t , t ), ⌃ 1 r,, is the covariance matrix of the parameters and := 2 3 (1 ↵ )isthequantileofthe chi-square distribution with three degrees of freedom. We should remark here that ellipsoidal specifications of uncertainty appear naturally in multivariate Gaussian settingsfortheuncertaintyaboutthedriftsoftradeableassetprices,andliterature can be referred to [8,10,34]. In [21], the pricing bound for the Heston call option 136 under model ambiguity is derived and proved to be the unique solutions of the following BSDEs with payo↵ g(S T )=(S T K) + at maturity T, 8 > > > < > > > : dY ± t = H ± (S t ,V t ,Y ± t ,Z t )dt+Z t dW t ,Y ± T =g(S T ), H ± (S t ,V t ,Y ± t ,Z t )=± p ⌘ T t ⌃ T ⌘ t rY t , (3.16) where ⌘ t 2R 3⇥ 1 is the vector of coecients to the parameter deviances of equation given by ⌘ t = " ( Z 2 t p 1 ⇢ 2 p V t Y t ),( Z 1 t p V t + ⇢Z 2 t p V t p 1 ⇢ 2 ),( Z 1 t p V t ⇢Z 2 t p 1 ⇢ 2 p V t ) # T , and Z t ,W t 2R 2 . Also, the perspective deviance towards the true parameter cor- responding to (3.16) are u ± (S t ,V t ,Y ± t ,Z t )= ± q ⌘ T t ⌃ T ⌘ t ⌃ ⌘ t .Followingtheidea in [21], the forward component X=(S,V)oftheSDEisgeneratedbystandard Euler-Maruyama scheme for the log-price and an implicit Milstein scheme for the variance 8 > > > < > > > : logS ⇡ t =logS ⇡ t i 1 +(r 1 2 V ⇡ t i 1 ) i + p V ⇡ t i 1 (⇢ W 1 t i + p 1 ⇢ 2 W 2 t i 1 ), V ⇡ t i = V ⇡ t i 1 +✓ i + p V ⇡ t i 1 W 2 t i + 1 4 2 (( W 2 t i ) 2 i ) 1+˜  i , 137 where W 1 t i , W 2 t i areindependentvariablesgeneratedfromthezero-meannormal distribution with variance i . High dimensional and non-linear example. We present the equivalent PPDE of our path-dependent BSDE example in (3.8). On the canonical space ([0,T]⇥ C([0,T],IR d )), the PPDE follows 8 > > > < > > > : @ t u+ 1 2 tr(@ !! u)+f(t,!,u,@ ! u)=0, u(T,!)= ⇠ (!). For the high dimensional example with generator f =0,andterminalcondition ⇠ (!)= ⇣ R T 0 P d i=1 ! i s ds ⌘ 2 ,thisPPDEyieldsanexplicitsolution u(t,!)= Z t 0 d X i=1 ! i s du ! 2 + d X i=1 ! i t ! 2 (T t) 2 +2(T t) d X i=1 ! i t ! Z t 0 d X i=1 ! i s ds+ d 3 (T t) 3 . For the non-linear example, the equivalent PPDE follows 8 > > > < > > > : @ t u+ 1 2 @ !! u+min µ2 [µ,µ] µ@ ! u+f 0 (t,!,¯ !)=0 u(T,!)=g(! T ,¯ ! T ), (3.17) 138 where g(! T ,¯ ! T )=cos(! T +¯ ! T ), and ¯ ! t := R t 0 ! s ds,and f 0 (t,!,!)=max(sin(! t +! t ),0)(µ+! t )+min(sin(! t +! t ),0)(µ+! t )+ 1 2 cos(! t +! t ). As for the BSDE, the terminal condition is given by Y T =cos(X T + R T 0 X s ds), and the forward asset process dX t = dB t . The solution is explicitly given by Y t =cos(X t + R t 0 X s ds). This example is inspired by a two person zero sum game from [66]. 3.5.3 Proof In this section, we study the universality approximation property of the Markov FBSDE(3.1)byusingthedeepsignatureandDNNinthestandardEulerschemes. As for the convergence analysis of the deep signature non-Markovian BSDE algo- rithm, we leave for future study. For notation simplicity, we may carry out the proofs only for one-dimensional case, i.e. d = d 1 = d 2 =1. Letusrecallthe following assumptions on the coecients. Assumption 3.5.5. Let the following assumptions be in force. • b,,f,g are deterministic taking values inR d1 , R d1⇥ d , R d2 , R d2 , respectively; and b(·,0), (·,0),f(·,0,0,0) and g(0) are bounded. • b,,f,g are smooth enough with respect to all variables (t,x,y,z) and all derivatives are bounded by constant L. 139 Before we show the main estimates in this section, we first introduce the fol- lowing universality property for neural network from [33], see also [49]. Lemma 3.5.6. Let ˆ (x) be a sigmoid function (i.e. a non-constant, increasing, and bounded continuous function on R). Let K be any compact subset of R n , and ˆ f :K ! R d be a continuous function mapping. Then for an arbitrary✏> 0, there exists an integerN> 0, an d⇥ N matrix A and an N dimensional vector ✓ such that max x2 K | ˆ f(x) Aˆ (Bx+✓ )| ✏, holds where ˆ :R N ! R N is a sigmoid mapping defined by ˆ ( 0 (u 1 ,···u N )) = 0 (ˆ (u 1 ),···, (u N )). For the time horizon [0,T], we denote h := (T t)/n as the step size for the standard Euler scheme, and we denote t i := ih,i=0,···,n. Similarly, for some k 2 R + ,wedenote u := k⇥ (T t)/n as the step size for the deep signature Euler scheme, and we denote u i := iu = ikh,with u i = u i u i 1 = kh,for i = 0,···,n/k :=e n.Furthermore,wekeeptheconventionthat W i+1 := W i+1 W i 140 and W u i+1 :=W u i+1 W u i . According to deep signature Euler scheme (3.4), we have Y ˜ n,Sig u i :=Y ˜ n,Sig u i 1 f(u i 1 ,X n u i 1 ,Y ˜ n,Sig u i 1 ,Z ✓, Sig u i 1 ) u i +Z ✓, Sig u i 1 W u i . where Z ✓, Sig u i := R ✓ i ({S u j } i j=1 ). At last, we denote Y n u i and Z n u i as values for the standard Euler scheme approximation ofY andZ (equation (3.1)) at timeu i .The following estimate is a standard result for Markov BSDEs, see [79][Theorem 5.3.3]. Lemma 3.5.7. Let Assumptions 3.2.1 hold and assume h is small enough. Then max 0 i n E h sup t i  t t i+1 |Y t Y t i | 2 i + n 1 X i=0 E h Z t i+1 t i |Z t Z t i | 2 dt i  C[1+|x| 2 ]h. With the above lemma in hand, we are ready to prove the universal approxi- mation property. Proof of Lemma 3.2.2. We assume that constant C changes generically from line to line. Applying the triangle inequality, for t2 [u i ,u i+1 ], we have |Z t Z ✓, Sig u i | 2  2[|Z t Z u i | 2 +|Z u i Z ✓, Sig u i | 2 ], which implies ˜ n 1 X i=0 E h Z u i+1 u i |Z t Z ✓, Sig u i | 2 dt i  2 n ˜ n 1 X i=0 E h Z u i+1 u i |Z t Z u i | 2 dt+ Z u i+1 u i |Z u i Z ✓, Sig u i | 2 dt io . 141 By Lemma (3.5.7), we can obtain that ˜ n 1 X i=0 E h Z u i+1 u i |Z t Z u i | 2 dt i  C(1+|x| 2 )kh. Furthermore, since ˜ n⇥ kh =T, we observe that ˜ n 1 X i=0 E h Z u i+1 u i |Z u i Z ✓, Sig u i | 2 dt i  T max 0 i ˜ n 1 E[|Z u i Z ✓, Sig u i | 2 ]. Now it suces to show that for any ✏,thereexistsnetwork ✓ such that max 0 i ˜ n 1 E[|Z u i Z ✓, Sig u i | 2 ]  ✏.Fromthenon-linearFeynman-Kacformula, we know Z t = @ x u (t,X t ):=F(t,X t ). For simplicity, we further assume that the solution u2C 1 ,1 b and 2C 1 b ,which implies thatF2C 1 ,1 b .WethushavethefollowingTaylorexpansion, dZ t =dF(t,X t )= @ t F(t,X t )dt+@ x F(t,X t ) dX t . (3.18) 142 Applying the change of variable formula iteratively, we get the following local approximation by using Taylor expansion at step N, Z t Z s =F(t,X t )F (s,X s )⇡ N X k=1 F k ( b X s ) Z I k d b X t 1 ⌦··· d b X t k where we denote I k :={s<t 1 <t 2 <···<t k <t} as the subdivision of the time interval [s,t], and { b X t } t2 [0,T] := {t,X t } t2 [0,T] as the enhanced path of the time parameter and the path {X t } t2 [0,T] .Thecoecientterm F k in the above Taylor expansion is defined recursively, F 1 =F =: @ x u, F k+1 =D(F k ), where D denotes the di↵erential operator. Following the idea in [49][Section 4], we consider the step-N Taylor expansion of Z,denotedas { b Z u i } ˜ n i=0 .Wehavethe following approximation of Z u i , Z u i =F(u i ,X u i )⇡ b Z u i = b Z u i 1 + N X k=1 F k ( b X u i 1 ) b X k u i 1 ,u i = g F N (l u i 1 , b Z u i 1 ), = e g F N (S u i 1 , b Z u i 1 ), 143 where l u i 1 is the log-signature layer of b X,and S u i 1 is the log-signature layer of b X. Plugging inZ ✓, Sig u i =R ✓ ((S k ) u i k=1 )(orZ ✓, LSig u i =R ✓ ((l k ) u i k=1 )), for anyu i ,wehave |Z u i Z ✓, Sig u i |  | Z u i b Z u i |+| b Z u i Z ✓, Sig u i |  | Z u i b Z u i |+|g F N (l u i 1 , b Z u i 1 ) Z ✓, Sig u i | or  | Z u i b Z u i |+|e g F N (S u i 1 , b Z u i 1 ) Z ✓, LSig u i | ApplyingLemma3.5.6, forany">0, aslongaskh =T/e nissmallenoughandthe truncation order of the signature is large enough, we can always find a ✓ such that max ˜ n 1 i=0 E[|Z u i Z ✓, Sig u i | 2 ] ".Inparticular, "is independent of time discretization. If we replace the signature with log-signature, a similar proof for forward SDE can be found in [49][Theorem 4.1]. Theorem 3.5.8. Let Assumption 3.2.1 be in force. Then one can conclude that max 0 i ˜ n E[sup u i  t u i+1 |Y t Y ˜ n,Sig u i | 2 ] C[1+|x| 2 ]kh. Proof. Applying triangle inequality, we have |Y t Y ˜ n,Sig u i | 2  C[|Y t Y u i | 2 +|Y u i Y ˜ n,Sig u i | 2 ]. 144 According to Lemma 3.5.7, one can obtain that max 0 i ˜ n E[sup u i  t u i+1 |Y t Y u i | 2 ] C(1+|x| 2 )kh. (3.19) Next, it suces to show that max 0 i ˜ n E[|Y u i Y n,Sig u i | 2 ]  C(1 +|x| 2 )kh.We denote Y ˜ n,Sig u i :=Y u i Y ˜ n,Sig u i ,and I u i t :=f(t,X t ,Y t ,Z t ) f(u i ,X u i ,Y ˜ n,Sig u i ,Z ✓, Sig u i ). We thus have Y ˜ n,Sig u i+1 = Y ˜ n,Sig u i Z u i+1 u i I u i t dt+ Z u i+1 u i Z t Z ✓, Sig u i dW t . Taking squares on both sides and taking expectation, we have E[| Y ˜ n,Sig u i+1 | 2 ] CE h | Y ˜ n,Sig u i | 2 + Z u i+1 u i |I u i t | 2 dt+ Z u i+1 u i |Z t Z ✓, Sig u i | 2 dt i . According to Assumption 3.2.1, we further conclude that |I u i t | 2  | f(t,X t ,Y t ,Z t ) f(u i ,X u i ,Y u i ,Z u i )| 2 +|f(u i ,X u i ,Y u i ,Z u i ) f(u i ,X u i ,Y ˜ n,Sig u i ,Z ✓, Sig u i )| 2  L[(kh) 2 +|Y t Y u i | 2 +|X t X u i | 2 +|Z t Z u i | 2 +| Y ˜ n,Sig u i | 2 +|Z u i Z ✓, Sig u i | 2 ]. (3.20) 145 Plugging (3.20) into previous estimates, we obtain E h Z u i+1 u i |I u i t | 2 dt i  E h L Z u i+1 u i |Y t Y u i | 2 +|X t X u i | 2 +|Z t Z u i | 2 dt +L(kh) 3 +Lkh| Y ˜ n,Sig u i | 2 +Lkh|Z u i Z ✓, Sig u i | 2 i . Applying Lemman 3.5.7 and Lemma 3.2.2, we further get the following estimates E[ Z u i+1 u i |I u i t | 2 dt]  C(1+|x| 2 )kh+L(kh) 2 +Lkh"+LkhE[| Y ˜ n,Sig u i | 2 ]  C(1+|x| 2 )kh+LkhE[| Y ˜ n,Sig u i | 2 ]. 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Creator Luo, Man (author)

Core Title Topics on dynamic limit order book and its related computation

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Publication Date 07/27/2021

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