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On non-zero-sum stochastic game problems with stopping times

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On non-zero-sum stochastic game problems with stopping times

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Content ON NON-ZERO-SUM STOCHASTIC GAME PROBLEMS WITH STOPPING TIMES by Jie Du A Dissertation Presented to the 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 2012 Copyright 2012 Jie Du Dedication To my family ii Acknowledgments I am deeply indebted to my advisor, Professor Jianfeng Zhang for persevering with me throughout the time it took me to complete this research and write the dissertation. I always feel lucky to have such a gentleman as my advisor. This dissertation is only possible under the guidance of his immense knowledge, great enthusiasm, exceptional ideas and ongoing encouragement. I greatly appreciate Professors Jin Ma, a respectable scholar and teacher who shared his wisdom and experience unreservedly. My thanks must as well go to Professor Peter Baxendale, Michael Magill, Remigijus Mikulevicius and Fernando Zapatero for serving my qualifying and/or dissertation committees and for all the comments and advices provided to me. It is my honor to be a member of USC math department. Thanks to the effort of all the faculties and staffs, I enjoyed studying here every day in the past five years. I must acknowledge many friends who assisted, advised, and supported me. Especially, I need to express my gratitude to Jianfu Chen, Huanhuan Wang and iii Xinyang Wang whose friendship, hospitality, knowledge and humor have enlight- ened and entertained me over the years. Lastbutnotleast,IwouldliketothankmyparentsMeiqiuHuangandXiaoping Du, my fianc´ ee Zhongxiao Yang and all my family members. My mother deserves special thanks for providing me the best education and environment to grow up. I dedicate this dissertation to my family. iv Table of Contents Dedication ii Acknowledgments iii List of Figures vii Abstract viii Chapter 1: Introduction 1 1.1 Motivations and Literature Reviews . . . . . . . . . . . . . . . . . . 1 1.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Chapter 2: Preliminaries 8 2.1 Probability Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Classical Results to BSDE and RBSDE . . . . . . . . . . . . . . . . 11 Chapter 3: Static Non-Zero-Sum Dynkin Game Problem 16 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Construction of a Nash Equilibrium Point . . . . . . . . . . . . . . 18 3.4 Recursive Utility via BSDE and RBSDE . . . . . . . . . . . . . . . 28 Chapter 4: Dynamic Non-Zero-Sum Dynkin Game Problem and System of RBSDEs 40 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Some Preparation Results . . . . . . . . . . . . . . . . . . . . . . . 44 4.3 Decomposition and Equivalency . . . . . . . . . . . . . . . . . . . . 51 4.4 Three Special Solutions to the Equivalent RBSDE System . . . . . 57 4.5 Discrete Time Stopping Framework . . . . . . . . . . . . . . . . . . 58 4.6 Semimartingale Framework. . . . . . . . . . . . . . . . . . . . . . . 64 v Chapter 5: Principle-Agent Problem on Stopping Times 71 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . 72 5.3 2-Period Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.4 3-Period Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.5 Multi-Period Problem . . . . . . . . . . . . . . . . . . . . . . . . . 83 Bibliography 90 vi List of Figures 4.1 An Example of Equilibrium Nonuniqueness . . . . . . . . . . . . . . 42 5.1 An Example of Time Inconsistency . . . . . . . . . . . . . . . . . . 75 vii Abstract This dissertation consists of three parts. We first study the continuous time non- zero-sum Dynkin game which is a multi-player non-cooperative game on stopping times. We show that the Dynkin game has a Nash equilibrium point for general stochastic processes. The study extends the result of Hamad` ene and Zhang [28]. The second part is to study the value dynamics of Dynkin game. In a zero- sum Dynkin game, where one player’s cost is the other player’s benefit, the value process is characterized by a two-barrier reflected backward stochastic differential equation, see Cvitani´ c and Karatzas [11]. We build a parallel result for non-zero- sum Dynkin game and propose a new form of equilibrium called time-autonomous, which is mainly used to overcome non-uniqueness of the equilibrium, see Exam- ple 4.1. Under this framework, we construct the equivalency relation between a reflected BSDE system with jumps and non-zero-sum Dynkin Game. FinallywestudyPrincipal-Agentproblemonstoppingtimes,inwhichExample 5.4 addresses time-inconsistent issue in the sense that Bellman’s principle does not viii hold. We propose a method to solve Principal-Agent problem in discrete time framework. ix Chapter 1 Introduction 1.1 Motivations and Literature Reviews According to Myerson [45], game theory is ”the study of mathematical models of conflict and cooperation between intelligent rational decision-makers”. It has attracted a lot of research attentions since Neumann published a paper in 1928 [48]. In 1950, Nash [47] developed a criterion for mutual consistency of players’ strategies, known as Nash equilibrium. In 1965, Selten [52] introduced his solu- tion concept of subgame perfect equilibria. In 1967, Harsanyi [29] developed the concepts of complete information and Bayesian games. Nash, Selten and Harsanyi became Economics Nobel Laureates in 1994 for their contributions to economic game theory. Games could be differentiated by their characteristics: Cooperative or Non- cooperative; Zero-sum or Non-zero-sum; Discrete or Continuous and etc. Provided with ample of variations, game theory has been applied to wide array of economic areas including auction, asset allocation, industrial organization and contract the- ory. 1 Dynkin games of zero-sum or non-zero-sum, continuous or discrete time types, are games on stopping times. Since their introduction by Dynkin in [16], a lot of associated research activities have been performed, see e.g. [1, 2, 3, 4, 6, 9, 11, 17, 18, 22, 25, 38, 39, 43, 44, 46, 50, 53, 55] and the references therein. Stopping times have the property that the decision to stop at time t must be based on information available at time t. In financial market, three main subjects are what, by how much and when to exchange assets. Stopping times as strategies care about the last question. Optimal stopping problem is concerned with the problem when to make an action to optimize utility. For example, pricing Amer- ican option is all about how to find optimal exercise time. In general, optimal stopping problems could be structured using Bellman’s equation, hence solved by dynamical programming principle. Neveu [49] discussed optimal stopping for ran- dom processes in continuous time and his result is well summarized in Karatzas and Shreve [33]. El Karoui, Kapoudjian, Pardoux, Peng and Quenez [21] studied one- dimensional Reflected Backward Stochastic Differential Equations (RBSDE for short) and established its wellposedness. Moreover, they revealed that the solu- tion can be associated with a classical deterministic Skorohod problem, and also showed that the solution of the RBSDE coincides with the value of an optimal stopping time problem. Since then, RBSDE becomes a powerful tool to handle optimal stopping problems. 2 Zero-sum Dynkin game involves two players, each of whom can decide to stop it at a stopping time of his choice; upon termination a certain amount, which is random and depends on the time of termination, is paid by one of the players to the other, i.e. one player’s benefit is identically the other player’s cost. It was intensively studied by Cvitani´ c and Karatzas [11]. They established existence and uniqueness results for adapted solutions of backward stochastic differential equa- tions with two reflecting barriers, which generalized El Karoui et al [21]. Cvitani´ c and Karatzas [11] also built the connection between RBSDE with two barriers and zero-sum Dynkin game. A pathwise approach to zero-sum Dynkin game was intro- duced due to this connection. Hamad` ene and Hassani [26] removed Mokobodzki’s condition and regularity of the barriers imposed in [11]. They proved the existence of the solution to RBSDE with two barriers as long as barriers are completely separated. There are several variations of zero-sum stochastic Dynkin games. For example, Karatzas and Sudderth [34] studied stochastic games incorporating with both controls and stopping. They assumed controls of both players enter into the dynamics of payoff process. Karatzas and Zamfirescu [35] took a martingale approach to the same problem. Their method provided pathwise characterization of saddle point in addition. Comparingwiththezero-sumsetting,therearemuchfewerresultsonnon-zero- sum Dynkin game in literature. Hamad` ene and Zhang [28] constructed the Nash equilibrium point for general stochastic processes under mild conditions. Their 3 results solved the game problem statically, in the sense that players determine stopping strategies once at the initial time, and then wait for the termination of the game. A natural question is what if players are actively managing to choose best stopping strategies as time goes by; how the value functions evolve through time. However, in non-zero-sum game there are no constraints on utility processes from which players gain, the static game may have more than one equilibrium, see Example 4.1. Then which equilibrium and value function should we use? To overcome the difficulty, we propose a new form of equilibrium called time- autonomous. Time-autonomous equilibrium confines the number of solutions to the reflected backward SDE system. We shall establish an equivalence result on non-zero-sum Dynkin game and the associated RBSDE system. We note that Li [40] and Karatzas and Li [31, 32] also proposed a RBSDE system whose solution corresponds to a non-zero-sum game with both control and stopping. However, their RBSDE system is somewhat different from ours and they didn’t establish the equivalency between their RBSDE system and the original game problem. Players involved in Dynkin game are symmetrical, in the sense that no one has dominating power than the others. The players are indifferent about switching their indices, and the switching of the position won’t cost or benefit any of them. One may connect Dynkin game with Cournot game [10] and Bertrand game [5]. This assumption is not always the case as what actually happened on the mar- ket. Players in the market have different resources and information which lead to 4 different perspectives and levels of power. Let us consider two parties, say a com- pany and a job seeker in an employment contract. It rarely happens that company and job seeker have the same bargaining power and encounter similar structure of obligations to the other party. More often, the company may propose a standard package to each job seeker to avoid additional operating cost, and the job seeker may have the right to refuse to accept or negotiate based on the standard package. This phenomenon motivates us to consider another type of optimal stopping game problem which called Principal-Agent problem. One may connect the Principal-Agent problem with Stackelberg leadership model [23]. The Stackelberg leadership model is a strategic game in economics in which the leader firm moves first and then the follower firm moves sequentially. Theproblemistypicallysolvedbytwostepsoptimization. SolvingPrincipal-Agent problemhasthesameflavorwherevalueisunique,whileatime-inconsistentissueis raised. Time-inconsistentissueformallymeanstheBellman’sequationanddynam- icalprogrammingprincipaldonotapply, seeExample5.4. Time-inconsistentissue has been long recognized, since the original publication of Mean-Variance analysis for optimal asset allocation was published by Markowitz [41] in 1952. Strotz [54] found that the solution to Mean-Variance optimal portfolio problem in a multi period framework is inconsistent through time. This phenomenon makes dynamic programming principal invalid, even the definition of optimal strategy is ambigu- ous. 5 RecentdevelopmentonmathematicaltreatmentofAssetAllocationandBehav- ioral Finance recalled time-inconsistent issue, see e.g. [7, 8, 24, 30, 37, 42, 51, 56, 57, 58] and references therein. Bj¨ ork and Mergoci [7] provided a general theory of Markovian time-inconsistent stochastic control problems. They indicated two basic ways of handling time inconsistency. One is pre-committed approach where investorschooseoptimalstrategyattimezeroandstickwiththeiroriginalcommit- ments. The other approach is to study time-inconsistent issue within game theo- reticframework. Gametheoreticapproachessentiallymodifiesthetargetfunctions of investors, hence transfers Principal-Agent problem to Dynkin game. What we use in this paper is the first approach, that is to study the static Principal-Agent problem since it is already interesting and difficult enough. We propose a method to solve discrete Principal-Agent problem using Lagrange Multipliers. 1.2 Outline of the Thesis Themaingoalofthisdissertationistostudytwotypesofstochasticgameproblems onstoppingtimes,DynkingameandPrincipal-Agentproblem. Thisdissertationis organized as follows. In Chapter 2, we introduce the probability set-up, notations, key assumptions and fundamental results to BSDEs and RBSDEs. In Chapeter 3, we introduce static Dynkin game set-up and study continuous time non-zero-sum Dynkingamewhichinvolvesmulti-playeraswellasrecursiveutility. Weshowthat the Dynkin game has a Nash equilibrium point for general stochastic processes. 6 The study extends the work Hamad` ene and Zhang [28]. Chapter 4 is devoted to study value dynamics of non-zero-sum Dynkin game. We construct an equiva- lent reflected backward SDE system which offers a new tool of studying dynamic non-zero-sum Dynkin game. Three special cases where reflected backward SDE system has a solution are provided, and we also try to regulate the circumstances where multiple Nash equilibrium points may appear. In Chapter 5, we introduce Principal-Agent problem on stopping times. Time-inconsistent issue is addressed, and we outline a method to solve Principal-Agent problem in discrete time frame- work. 7 Chapter 2 Preliminaries 2.1 Probability Set-up Let [0,T] be a finite time interval and let (Ω,F,P,F,B) be a standard set-up: (Ω,F,P) is a complete probability space;F, {F t } t∈[0;T] is a filtration satisfying the usual hypotheses i.e. it is complete and right continues.; and B is anF Brow- nian motion. Particularly, we say the standard set-up is Brownian if F t = F B t , the natural filtration generated by the Brownian motion B, augmented by all the P−null sets of F and satisfying the usual hypotheses. 2.2 Notations Throughout the paper, we used the following notations. • T is a real positive constant which stands for the horizon of the problem and (Ω,F,P) is a fixed probability space on which is defined a filtration F △ = (F t ) 0≤t≤T which satisfies the usual hypotheses. 8 • For anyF-stopping time θ, let T denote the set ofF-stopping time τ such that τ ∈ [θ,T], P −a.s. • Let D denote the space ofF-adaptedR-valued right continuous with left limits (RCLL for short) processes X such that the set of random variables {X ,τ ∈T 0 } are uniformly integrable. • We adopt the notations of Hamad` ene and Hassani [26] (or Cvitani´ c and Karatzas [11]): S ci (resp. S 2 ci ) the set of continuous P-measurable non- decreasing processes K △ ={K t } t≤T such that K 0 = 0 (resp. and E[(K T ) 2 ] < ∞). • S 2 the set of P-measurable and continuous processes ω ={ω t } t≤T such that E[sup t≤T |ω t | 2 ]<∞. • Consider a Dynkin game problem with n players a i , i = 1,2··· and n, who control their own stopping times τ i , i = 1,2,··· and n. Denote a strategy as τ e △ = (τ 1 ,τ 2 ,··· ,τ n ); and a strategy less player i’s as τ −i f △ = (τ 1 ,τ 2 ,··· ,τ i−1 ,τ i+1 ,··· ,τ n ). Similarly, a strategy for Principal-Agent prob- lem (two players in total) is denoted as (τ,σ). • Eachplayera i isassociatedwithtwopayoff/costprocessesL i (thelowerone), U i (the higher one). The i th player’s utility is J i (τ e ) =J i (τ i ;τ −i f ) △ =E{L i i 1 { i < i } +U i i 1 { i ≤ i } },∀i∈{1,2,··· ,n} (2.1) 9 where δ i △ = min j̸=i {τ j }. That is, if the player a i is the one who essentially stops the game (i.e. τ i <τ j for all j ̸=i), then he receives L i i ; if the game is stopped by other players, i.e. δ i ≤τ i , then a i receives U i i instead. 2.3 Assumptions Throughout Chapter 3 and 4, we shall use the following assumptions. Assumption 2.1. The processes L i ,U i ∈ D, L i have only positive jumps and L i t ≤U i t , ∀t≤T, P−a.s. ∀i = 1,2,··· ,n. Consider the following nonlinear BSDE: Y t =ξ + ∫ T t f(ω,s,Y s ,Z s )ds− ∫ T t Z s dB s (2.2) where f : Ω×[0,T]×R×R→R is progressively measurable and f(·,·,y,z)∈F for any y,z. ForF−adapted processes Y,Z, denote ||(Y,Z)|| 2 △ =E{ sup 0≤t≤T |Y t | 2 + ∫ T 0 |Z t | 2 dt} (2.3) It is necessary to impose some standard assumptions to make the above BSDE system have solutions. 10 Assumption 2.2. (i)F =F B (ii) ξ ∈L 2 (F T ) and f(·,0,0)∈L 2 (F) (iii) f is uniformly Lipschitz continuous in (y,z). That is, |f(t,y 1 ,z 1 )−f(t,y 2 ,z 2 )|≤C[|y 1 −y 2 |+|z 1 −z 2 |], ∀(ω,t,y i ,z i ) i = 1,2 Assumption 2.3. Two processes U △ ={U t } t≤T and L △ ={L t } t≤T belong to S 2 and satisfy L t ≤U t , ∀t≤T, and L T ≤ξ ≤U T . 2.4 Classical Results to BSDE and RBSDE With Assumption 2.2, we cite three classical results in BSDE: Existence, Stability andComparisonwhichareextremelyimportanttotheanalysisthroughthispaper. FortheirproofswerefertoElKaroui,PengandQuenez[20]. ConsiderBSDE(2.2), Theorem 2.4. (BSDE Existence, [20]) Assume Assumption 2.2. Then BSDE (2.2) has a unique solution. Theorem 2.5. (BSDE Stability, [20]) Assume Y t =ξ + ∫ T t f(ω,s,Y s ,Z s )ds− ∫ T t Z s dB s 11 and Y n t =ξ n + ∫ T t f n (ω,s,Y n s ,Z n s )ds− ∫ T t Z n s dB s , n = 1,2,··· where (ξ,f) and (ξ n ,f n ) satisfy Assumption 2.2 uniformly. Assume further that lim n→∞ E{|ξ n −ξ| 2 + ∫ T 0 |f n (t,0,0)−f(t,0,0)| 2 dt} = 0 and ∀(t,y,z), f n (t,y,z)→f(t,y,z), P −a.s. Then lim n→∞ ||(Y n −Y,Z n −Z)|| = 0 Theorem2.6. (BSDE Comparison, [20]) Assume Assumption 2.2, for (ξ i ,f i ),i = 1,2. Let (Y i ,Z i ) be the solution to the following BSDE Y i t =ξ + ∫ T t f i (ω,s,Y i s ,Z i s )ds− ∫ T t Z i s dB s , i = 1,2. Assume further thatξ 1 ≥ξ 2 P−a.s. and∀(t,y,z), f 1 (t,y,z)≥f 2 (t,y,z) P−a.s. Then Y 1 t ≥Y 2 t , P −a.s. ∀t. In particular, Y 1 0 ≥Y 2 0 . 12 We introduce RBSDE to help constructing optimal stopping times. Adopt Assumption 2.2 and 2.3 to guarantee the existence of solution to RBSDE. Here is the definition of RBSDE,                Y t =ξ + ∫ T t f(s,Y s ,Z s )ds− ∫ T t Z s dB s +K T −K t Y t ≥L t (Y t −L t )dK t = 0 (2.4) Now we introduce some properties of RBSDE, whose proofs can be found in El Karoui et al [21]. Consider RBSDE (2.4), Theorem 2.7. (RBSDE Existence, [21]) Assume Assumption 2.2 and 2.3. Then RBSDE (2.4) has a unique solution. Theorem 2.8. (RBSDE Comparison, Theorem 4.1, [21]) Let (ξ,f,L) and (ξ ′ ,f ′ ,L ′ ) be two sets of data, satisfying Assumption 2.2 and 2.3, and suppose in addition the following: (i) ξ ≤ξ ′ a.s. (ii) f(t,y,z)≤f ′ (t,y,z) dP ×dt a.e. (iii) L t ≤L ′ t ,0≤t≤T P −a.s. Let (Y,Z,K) be a solution of the RBSDE with data (ξ,f,L), and (Y ′ ,Z ′ ,K ′ ) be a solution of the RBSDE with data (ξ ′ ,f ′ ,L ′ ). Then Y t ≤Y ′ t , 0≤t≤T P −a.s. 13 Theorem 2.9. (RBSDE and optimal stopping time, [21]). Assume Assumption 2.2 and 2.3. If (Y,Z,K) is a solution to (2.4), then (i) K T −K t = sup t≤s≤T (ξ + ∫ T s f(r,Y r ,Z r )dr− ∫ T s Z r dW r −L s ) − (ii) Y t = esssup ∈Tt E t { ∫ t f(s,Y s ,Z s )ds+L 1 {<T} +ξ1 {=T} } (iii) The stopping time τ t △ = inf{s≥t :Y s =L s }∧T is optimal for (ii). When f ≡ 0 , Theorem 2.9 degenerates into Snell envelope Theorem 2.10 as follows. The Snell envelope, used in stochastic and mathematical finance, is the smallest supermartingale dominating a stochastic process. We will construct optimal stopping time τ ∗ e later based on the Snell envelope of processes which we introduce briefly now for readers convenience. For more details on this subject one can refer e.g. to El Karoui [19] or Dellacherie and Meyer [15]. Lemma 2.10. ([15], pp.431 or [19], pp.140) For any V ∈ D, there exists aF- adaptedR-valued RCLL process W such that W is the smallest supermartingale which dominatesV, i.e. if ¯ W is another RCLL supermartingale such that ¯ W t ≥V t for all 0 ≤ t≤ T, then ¯ W t ≥ W t for any 0 ≤ t≤ T. The process W is called the Snell envelope of V. Moreover, the following properties hold: (i) For anyF-stopping time θ we have: W = esssup ∈T E[V |F ] (and then W T =V T ), P −a.s. (2.5) 14 (ii) Assume that V has only positive jumps. Then the stopping time τ ∗ △ = inf{s≥ 0,W s =V s }∧T is optimal, i.e. E[W 0 ] =E[W ] =E[V ] = sup ∈T 0 E[V ]. (2.6) Remark 2.11. As a by-product of (2.6) we have W =V and the process W is a martingale on the time interval [0,τ ∗ ]. 15 Chapter 3 Static Non-Zero-Sum Dynkin Game Problem 3.1 Introduction To begin with let us briefly describe those game problems. Assume we have a system controlled by two players a 1 and a 2 . The system works or is alive up to the time when one of the players decides to stop the control at a stopping time τ 1 for a 1 and τ 2 for a 2 . An example of that system is a recallable option in a financial market (see [25, 36] for more details). When the system is stopped the payment for a 1 (resp. a 2 ) amounts to a quantity J 1 (τ 1 ,τ 2 ) (resp. J 2 (τ 1 ,τ 2 )) which could be negative and then it is a cost. We say that the non-zero-sum Dynkin game associated with J 1 and J 2 has a Nash equilibrium point (NEP for short) if there exists a pair of stopping times (τ ∗ 1 ,τ ∗ 2 ) such that for any (τ 1 ,τ 2 ) we have: J 1 (τ ∗ 1 ,τ ∗ 2 )≥J 1 (τ 1 ,τ ∗ 2 ) and J 2 (τ ∗ 1 ,τ ∗ 2 )≥J 2 (τ ∗ 1 ,τ 2 ). 16 The particular case where J 1 +J 2 = 0 corresponds to the zero-sum Dynkin game. In this case, when the pair (τ ∗ 1 ,τ ∗ 2 ) exists it satisfies J 1 (τ 1 ,τ ∗ 2 )≤J 1 (τ ∗ 1 ,τ ∗ 2 )≤J 1 (τ ∗ 1 ,τ 2 ), for any τ 1 , τ 2 . We call such a (τ ∗ 1 ,τ ∗ 2 ) a saddle-point for the game. Additionally this existence implies in particular that: inf 2 sup 1 J 1 (τ 1 ,τ 2 ) = sup 1 inf 2 J 1 (τ 1 ,τ 2 ), i.e. the game has a value. Hamad` ene and Zhang [28] studied the non-zero-sum game in continuous time under mild assumptions. They constructed a Nash equilibrium point (NEP for short) for two player game problem by using Snell Envelope technique. So it is naturaltoconsiderDynkinGamewithmulti-player. Byassumingsomesymmetric properties among players, we remove the assumption A3 in [28] and extend the result. We notice recently that Hamad` ene and Hassani [27] has obtained very similar results independently. ElKarouietal[21]revealedthatastochasticoptimalstoppingcontrolproblem could be solved by RBSDE. Since then, the theory of BSDE and RBSDE has become a powerful tool to attack problems related to optimal stopping time. In 17 laterpartofthechapter,weincorporaterecursiveutilityintonon-zero-sumDynkin game and solve it by using RBSDE. 3.2 Main Results We give a precise definition of Nash equilibrium point. Denition 3.1. We say that τ e ∗ ∈ T n 0 is a Nash Equilibrium Point of the non- zero-sum Dynkin game associated with {J i } i=1;2;···;n if: J i (τ i ;τ −i f ∗ )≤J i (τ ∗ i ;τ −i f ∗ ), ∀τ i ∈T 0 . (3.1) Theorem 3.2 and 3.14 are two main theorems in Chapter 3. Here is our first result which confirms the existence of the NEP with multi-player. Theorem 3.2. Under Assumption 2.1, the non-zero-sum Dynkin game associated with {J i } i=1;2;···;n has a NEP τ ∗ e = (τ ∗ 1 ,τ ∗ 2 ,··· ,τ ∗ n ). 3.3 Construction of a Nash Equilibrium Point Inthissectionweshallconstructasequenceoffamiliesofdecreasingstoppingtimes (τ mn+1 ,τ mn+2 ,··· ,τ mn+n ) and show that their limit (τ ∗ 1 ,τ ∗ 2 ,··· ,τ ∗ n ) is a NEP. 18 We start with defining τ 1 △ = T,τ 2 △ = T,··· ,τ n △ = T. For k = 0,1,···, assume τ k+1 ,τ k+2 ,··· ,τ k+n have been defined, we then define τ k+n+1 as follows. First, let W k+n+1 t △ = esssup ∈Tt E t {L (k+1) 1 {< n+k } +U (k+1) n+k 1 { n+k ≤} }, t≤T; (3.2) where and in the sequel E t {·} △ =E{·|F t } and σ n+k △ = min{τ n+k ,τ n+k−1 ,··· ,τ k+2 }, (·) is in the (mod n) sense, i.e. L (k) △ =L i , U (k) △ =U i , if k≡i (mod n). Define ˜ τ n+k+1 △ = inf{t≥ 0 :W n+k+1 t =L (n+k+1) t }∧σ n+k (3.3) τ n+k+1 △ =        ˜ τ n+k+1 if ˜ τ n+k+1 <σ n+k τ k+1 if ˜ τ n+k+1 =σ n+k (3.4) It is easily to see that {W,˜ τ,τ,δ} are well defined recursively. The next lemma reveals the nature of {τ k } k=1;2;··· , and it is a core ingredient of Theorem 3.2. Lemma 3.3. Assume Assumption 2.1 . For k = 1,2,··· τ k is a stopping time and τ k+n ≤τ k . proof. We prove the following stronger results by induction on k: (1) τ k ∈T 0 (2) {τ k <σ n+k−1 }⊂{˜ τ n+k ≤τ k } (3) τ n+k ≤τ k ∀k∈N (3.5) 19 Obviously (3.5) holds for k = 1,2,··· ,n. Assume it is true for k = mn−n+ 1,mn−n+2,··· ,mn. We shall prove it for k =mn+1,mn+2,··· ,mn+n. Define V n+k+1 t △ =L (k+1) t 1 {t< n+k } +U (k+1) n+k 1 { n+k ≤t} (3.6) for k = 0,1,··· ,mn−n. Since τ k+2 ,τ k+3 ,··· ,τ n+k are stopping times, by Assumption 2.1 we know σ n+k ∈ T 0 , hence V n+k+1 ∈ D and it has only positive jumps. Apply Lemma 2.10, W n+k+1 is the snell envelope of V n+k+1 and ˜ τ n+k+1 is the minimal optimal stopping time. To prove (3.5) (1), τ mn+1 ∈T 0 , we note that {˜ τ mn+1 =σ mn }⊂{τ (m−1)n+1 ≥σ mn } (3.7) By definition, τ mn+1 =        ˜ τ mn+1 ˜ τ mn+1 <σ mn τ (m−1)n+1 ˜ τ mn+1 =σ mn 20 In fact, if τ mn−n+1 <σ mn , then by the second claim of (3.5) for k =mn−n+1 we have ˜ τ mn+1 ≤ τ mn−n+1 . This implies ˜ τ mn+1 < σ mn and then (3.7) follows immediately. For any t≤T we have {τ mn+1 ≤t} = [ {˜ τ mn+1 ≤t}∩{˜ τ mn+1 <σ mn } ] ∪ [ {τ mn−n+1 ≤t}∩{˜ τ mn+1 =σ mn } ] = [ {˜ τ mn+1 ≤t}∩{˜ τ mn+1 <σ mn ≤t} ] ∪ [ {˜ τ mn+1 ≤t}∩{σ mn >t} ] ∪ [ {τ mn−n+1 ≤t}∩{˜ τ mn+1 =σ mn ≤t} ] Then {τ mn+1 ≤t}∈F t for any t and thus τ mn+1 is a stopping time. To prove (3.5) (2). We want to show that W n+k+1 k+1 1 { k+1 < n+k } =L (k+1) k+1 1 { k+1 < n+k } (3.8) for k =mn. By the definition of ˜ τ mn+1 in (3.3), This means (3.5) (2), i.e. {τ k+1 <σ n+k }⊂{˜ τ n+k+1 <τ k+1 } 21 On {τ k+1 < σ n+k }, the structure could be well explored. First, τ k+1 < min{τ k+2 ,τ k+3 ,··· ,τ k+n }. Then τ k+2 △ = τ k+2−n , since ˜ τ k+2 = σ k+1 . Otherwise ˜ τ k+2 <σ k+1 implies that σ n+k ≤τ k+2 △ = ˜ τ k+2 <σ k+1 ≤τ k+1 , which is a contradiction! Likewise, we could obtain                τ k+3 △ =τ k+3−n , ˜ τ k+3 =σ k+2 . . . τ k+n △ =τ k , ˜ τ k+n =σ k which together implies σ n+k =σ k . Thus W n+k+1 k+1 1 { k+1 < n+k } =W k+1 k+1 1 { k+1 < n+k } (3.9) On the other hand, if ˜ τ k+1 = σ k , by the third claim of (3.5) for k ≤ mn−n, (3.7), and definition of (3.3), we have τ k+1 = τ k−n+1 ≥ σ k ≥ σ n+k . Thus {τ k+1 < σ n+k }⊂{τ k+1 = ˜ τ k+1 <σ k }. Therefore by Remark 2.11, W n+k+1 k+1 1 { k+1 < n+k } =X (k+1) k+1 1 { k+1 < n+k } (3.10) 22 which together with (3.9) implies that (3.8), hence 3.5 (2). {τ mn+1 <σ mn+n }⊂{˜ τ mn+n+1 ≤τ mn+1 } (3.11) To prove (3.5) (3), if τ mn+1 < τ mn+n+1 , then τ mn+1 < τ mn+n+1 = ˜ τ mn+n+1 < σ mn+n . This is contradicts with (3.11). Therefore, τ mn+n+1 ≤τ mn+1 . Finally, one can prove (3.5) for k =mn+2,mn+3,··· ,mn+n similarly. By induction, we confirms Lemma 3.1 so far. Next lemma shows that τ k is an optimal stopping time for some approximating problem. Lemma 3.4. Assume Assumption 2.1. For any τ ∈T 0 and any k we have: J (k) (τ;τ k−1 ,τ k−2 ,··· ,τ k−n+1 )≤J (k) (τ k ;τ k−1 ,τ k−2 ,··· ,τ k−n+1 ) (3.12) proof. First, by the definition of W k+1 in (3.2) we have W k+1 k = U k+1 k . Next, by Lemma 2.10 we have W k+1 t = L k+1 t for any t ∈ [0,σ k ] and W k+1 is a super- martingale over [0,σ k ]. Then, for any τ ∈T 0 , J (k+1) (τ;τ k ,τ k−1 ,··· ,τ k−n+2 ) = E{L (k+1) 1 {< k } +U (k+1) k 1 { k ≤} } ≤ E{W k+1 1 {< k } +W k+1 k 1 { k ≤} } = E{W k+1 k ∧ }≤W k+1 0 (3.13) 23 On the other hand, J (k+1) (τ k+1 ;τ k ,τ k−1 ,··· ,τ k−n+2 ) = E{L (k+1) k+1 1 { k+1 < k } +U (k+1) k 1 { k ≤ k+1 } } = E{L (k+1) k+1 1 {˜ k+1 < k ; k+1 < k } +L (k+1) k+1 1 {˜ k+1 ≥ k ; k+1 < k } +U (k+1) k 1 {˜ k+1 = k } } (3.14) By (3.3), (3.7) and Remark 2.11, we get J (k+1) (τ k+1 ;τ k ,τ k−1 ,··· ,τ k−n+2 ) = E{L (k+1) k+1 1 {˜ k+1 < k } +0+W (k+1) k 1 {˜ k+1 = k } } = E{W (k+1) ˜ k+1 } =W k+1 0 (3.15) This, together with (3.13), proves the lemma. Now define τ ∗ i △ = lim m→∞ τ mn+i (3.16) We shall prove that {τ ∗ e } is a NEP. We divide the proof into several lemmas. Lemma 3.5. Assume Assumption 2.1. For any τ ∈ T 0 such that P(τ = σ ∗ −i < T) = 0, we have lim m→∞ J i (τ;τ mn+i−1 ^ ) =J i (τ;τ ∗ −i f ) proof. Since {τ <σ mn+i−1 }⊂{τ <T}, we have lim m→∞ E{L i 1 {< mn+i1 } } = lim m→∞ E{L i 1 {< mn+i1 ;̸= i } } =E{L i 1 {< i } } 24 Moreover, lim m→∞ E{U i mn+i1 1 { mn+i1 ≤} } = lim m→∞ E { U i i 1 { mn+i1 ≤} {1 {̸= i } +1 {= i } } } = E { U i i {1 {> i } +1 {= i =T} } } = E{U i i 1 {≥ i } }. (3.17) Then, lim m→∞ J i (τ;τ mn+i−1 ^ ) = lim m→∞ E{L i 1 {< mn+i1 } +U i mn+i1 1 { mn+i1 ≤} } = E{L i 1 {< i } +U i i 1 {≥ i } } = J i (τ;τ ∗ −i f ). (3.18) The proof is complete so far. Lemma 3.6. Assume Assumption 2.1. Then it holds that lim m→∞ J i (τ mn+i ;τ mn+i−1 ^ ) =J i (τ ∗ i ;τ ∗ −i f ), ∀i = 1,2,··· ,n. (3.19) 25 proof. First we have lim m→∞ J i (τ mn+i ;τ mn+i−1 ^ ) = lim m→∞ E { [L i mn+i 1 { mn+i < mn+i1 } +U i mn+i1 1 { mn+i ≥ mn+i1 } ] ×{1 { i ̸= i } +1 { i = i } } } = lim m→∞ E { L i i 1 { i < i } +U i i 1 { i > i } +[L i i 1 { mn+i < mn+i1 } +U i i 1 { mn+i ≥ mn+i1 } ]1 { i = i } } = E{L i i 1 { i < i } +U i i 1 { i ≥ i } }+I = J i (τ ∗ i ;τ ∗ −i f )+I (3.20) where I △ = lim m→∞ E { [L i i −U i i ]1 { i = i ; mn+i < mn+i1 } } . (3.21) On the other hand, set τ △ =        τ ∗ i τ ∗ i <σ ∗ −i T τ ∗ i ≥σ ∗ −i 26 Then τ ∈T 0 and P(τ =σ ∗ −i <T) = 0. By Lemma 3.5 we have lim m→∞ J i (τ;τ mn+i−1 ^ ) = J i (τ;τ ∗ −i f ) = E{L i 1 {< i } +U i i 1 {≥ i } } = E{L i i 1 { i < i } +U i i 1 { i ≥ i } } = J i (τ ∗ i ;τ ∗ −i f ) (3.22) By Lemma 3.4, we get I ≥ 0. Now by Assumption 2.1 we have I = 0. (3.23) Then (3.20) implies (3.19). We are now ready to show that τ ∗ e is a NEP. Proof of Theorem 3.2. First, by Lemma 3.5, Lemma 3.6, and Lemma 3.4, we have for any τ such that P(τ =σ ∗ −i <T) = 0, we have J i (τ;τ ∗ −i f )≤J i (τ ∗ i ;τ ∗ −i f ), ∀i = 1,2,··· ,n, τ ∈T 0 . (3.24) In the general case, for any τ ∈T 0 , set ˆ τ n △ =        [τ + 1 n ]∧T, if τ =σ ∗ −i <T; τ, otherwsie. 27 Then ˆ τ n is a stopping time and P(ˆ τ n =σ ∗ −i <T) = 0. Thus (3.24) leads to J i (τ;τ ∗ −i f )≤J i (τ ∗ i ;τ ∗ −i f ). Send n→∞, we obtain (3.24) for general τ. Thus we confirms τ ∗ e is a NEP. 3.4 Recursive Utility via BSDE and RBSDE Recursive utilities were first introduced by Duffie and Epstein [13, 14], indepen- dent of the development of the BSDE theory. It turns out that BSDE provides the perfect tool for such problems. In this section we study the non-zero sum game problem under recursive utilities, or more generally, our utility is defined via BSDEs. Forsimplicity,assumethereareonlytwoplayers,andwecouldformulatepayoff to each player as follows. J 1 (τ,σ) = y 1;; 0 J 2 (τ,σ) = y 2;; 0 (3.25) where, y 1 and y 2 satisfy BSDE, 28 y 1;; t = L 1 1 (<) +U 1 1 (≤) + ∫ ∧ t f 1 (s,y 1;; s ,z 1;; s )ds− ∫ ∧ t z 1;; s dB s y 2;; t = L 2 1 (<) +U 2 1 (≤) + ∫ ∧ t f 2 (s,y 2;; s ,z 2;; s )ds− ∫ ∧ t z 2;; s dB s (3.26) Remark 3.7. The recursive utility introduced by Duffie and Epstein [13, 14] cor- responds to the case that f is independent of z. In particular, when f = 0, the problem is reduced to the one in previous sections. Denition 3.8. We say that (τ ∗ 1 ,τ ∗ 2 )∈T 0 ×T 0 is a Nash Equilibrium Point of the non-zero-sum Dynkin game associated with J 1 and J 2 if: J 1 (τ 1 ,τ ∗ 2 )≤J 1 (τ ∗ 1 ,τ ∗ 2 ), J 2 (τ ∗ 1 ,τ 2 )≤J 2 (τ ∗ 1 ,τ ∗ 2 ), ∀τ 1 ,τ 2 ∈T 0 . (3.27) We start with defining τ 1 △ = T and τ 2 △ = T. For n = 1,···, assume τ 2n−1 and τ 2n have been defined, we then define τ 2n+1 and τ 2n+2 recursively. First, denote Y 1; 2n t as the solution of the following RBSDE,                Y 1; 2n t △ =U 1 2n + ∫ 2n t f 1 (s,Y 1; 2n s ,Z 1; 2n s )ds− ∫ 2n t Z 1; 2n s dB s +K 2n 2n −K 2n t Y 1; 2n t ≥L 1 t (Y 1; 2n t −L 1 t )dK 2n t = 0 (3.28) 29 and define the intermediate stopping time, ˜ τ 2n+1 △ = inf{t≥ 0 :Y 1; 2n t =L 1 t }∧τ 2n (3.29) τ 2n+1 △ =        ˜ τ 2n+1 , if ˜ τ 2n+1 <τ 2n ; τ 2n−1 , if ˜ τ 2n+1 =τ 2n . (3.30) Next, let                        Y 2; 2n+1 t △ =U 2 2n+1 + ∫ 2n+1 t f 2 (s,Y 2; 2n+1 s ,Z 2; 2n+1 s )ds− ∫ 2n+1 t Z 2; 2n+1 s dB s +K 2n+1 2n+1 −K 2n+1 t Y 2; 2n+1 t ≥L 2 t (Y 2; 2n+1 t −L 2 t )dK 2n+1 t = 0 (3.31) and, ˜ τ 2n+2 △ = inf{t≥ 0 :Y 2; 2n+1 t =L 2 t }∧τ 2n+1 (3.32) τ 2n+2 △ =        ˜ τ 2n+2 , if ˜ τ 2n+2 <τ 2n+1 ; τ 2n , if ˜ τ 2n+2 =τ 2n+1 . (3.33) Then by Theorem 2.9, Y 1; 2n t △ = esssup ∈Tt y 1;; 2n t Y 2; 2n+1 t △ = esssup ∈Tt y 2; 2n+1 ; t 30 i.e. Y 1; 2n t is the snell envelope of y 1;; 2n t and Y 2; 2n+1 t is the snell envelope of y 2; 2n+1 ; t . Lemma 3.9. Assume Assumption 2.1, 2.2 and 2.3. For n = 1,2,···, τ n is a stopping time and τ n+2 ≤τ n . proof. We prove the following stronger results by induction on n: τ n ∈T 0 , {τ n <τ n+1 }⊂{˜ τ n+2 ≤τ n }, τ n+2 ≤τ n . (3.34) Obviously (3.34) holds for n = 1,2. Assume it is true for 2n−1 and 2n. We shall prove it for 2n+1 and 2n+2. First, notice the definition of (3.26). Since τ 2n is a stopping time, by Assump- tion 2.1, 2.2 and 2.3, we know y 1;; 2n is well defined and it is continuous. Apply Theorem2.9,Y 1; 2n isthesnellenvelopeofy 1;; 2n and ˜ τ 2n+1 istheminimaloptimal stopping time. To prove τ 2n+1 ∈T 0 , we note that {˜ τ 2n+1 =τ 2n }⊂{τ 2n−1 ≥τ 2n }. (3.35) 31 In fact if τ 2n−1 < τ 2n , then by the second claim of (3.34) for 2n− 1 we have ˜ τ 2n+1 ≤τ 2n−1 . This implies ˜ τ 2n+1 <τ 2n and then (3.35) follows immediately. Now recall definition (3.29), for any t≤T we have {τ 2n+1 ≤t} = [ {˜ τ 2n+1 ≤t}∩{˜ τ 2n+1 <τ 2n } ] ∪ [ {τ 2n−1 ≤t}∩{˜ τ 2n+1 =τ 2n } ] = [ {˜ τ 2n+1 ≤t}∩{˜ τ 2n+1 <τ 2n ≤t} ] ∪ [ {˜ τ 2n+1 ≤t}∩{τ 2n >t} ] ∪ [ {τ 2n−1 ≤t}∩{˜ τ 2n+1 =τ 2n ≤t} ] . Then {τ 2n+1 ≤t}∈F t for any t and thus τ 2n+1 is a stopping time. Next, on {τ 2n+1 < τ 2n+2 }, by definition of τ 2n+2 in (3.32) we have τ 2n+2 = τ 2n . Then y 1;; 2n+2 t =y 1;; 2n t for t≥τ 2n+1 and thus Y 1; 2n+2 2n+1 1 { 2n+1 < 2n+2 } =Y 1; 2n 2n+1 1 { 2n+1 < 2n+2 } . (3.36) On the other hand, if ˜ τ 2n+1 = τ 2n , by the third claim of (3.34) for 2n, (3.35), and definition of (3.29), we have τ 2n+2 ≤ τ 2n ≤ τ 2n−1 = τ 2n+1 . Thus {τ 2n+1 < τ 2n+2 }⊂{τ 2n+1 = ˜ τ 2n+1 <τ 2n }, and therefore, by Theorem 2.9, Y 1; 2n 2n+1 1 { 2n+1 < 2n+2 } =L 1 2n+1 1 { 2n+1 < 2n+2 } . 32 together with (3.36), implies that Y 1; 2n+2 2n+1 1 { 2n+1 < 2n+2 } =L 1 2n+1 1 { 2n+1 < 2n+2 } . Now by the definition of ˜ τ 2n+3 in (3.29) we know {τ 2n+1 <τ 2n+2 }⊂{˜ τ 2n+3 ≤τ 2n+1 }. (3.37) Moreover, if τ 2n+3 > τ 2n+1 , by definition (3.29) we have τ 2n+3 = ˜ τ 2n+3 < τ 2n+2 . Then τ 2n+1 < ˜ τ 2n+3 < τ 2n+2 . This contradicts with (3.37). Therefore, τ 2n+3 ≤ τ 2n+1 . Finally, one can prove (3.34) for 2n + 2 similarly. This finishes the proof of Lemma 3.9. The following is another important property of the stopping times τ n . Lemma 3.10. Assume Assumption 2.1, 2.2 and 2.3. On {τ n = τ n−1 }, we have τ m =T for all m≤n. proof. The result is obvious for n = 2. Assume it is true for n, consider case when n+1, on {τ n+1 = τ n }, by the definition of τ n+1 in (3.29) or (3.32) we have τ n+1 =τ n−1 . Then τ n =τ n−1 and thus by induction assumption we get the result. 33 Next lemma shows that τ n is an optimal stopping time for some approximating problem. Lemma 3.11. Assume Assumption 2.1, 2.2 and 2.3. For any τ ∈ T 0 and any n we have: J 1 (τ,τ 2n )≤J 1 (τ 2n+1 ,τ 2n ) and J 2 (τ 2n+1 ,τ)≤J 2 (τ 2n+1 ,τ 2n+2 ). (3.38) proof. We need to use the comparison Theorem 2.6. As t ≤ τ ∧τ 2n ∧τ 2n+1 , compare the terminal payoff in the following BSDEs. y 1;; 2n t = y 1;; 2n ∧ 2n ∧ 2n+1 + ∫ ∧ 2n ∧ 2n+1 t f 1 (s,y 1;; 2n s ,z 1;; 2n s )ds − ∫ ∧ 2n ∧ 2n+1 t z 1;; 2n s dB s y 1; 2n+1 ; 2n t = y 1; 2n+1 ; 2n ∧ 2n ∧ 2n+1 + ∫ ∧ 2n ∧ 2n+1 t f 1 (s,y 1; 2n+1 ; 2n s ,z 1; 2n+1 ; 2n s )ds − ∫ ∧ 2n ∧ 2n+1 t z 1; 2n+1 ; 2n s dB s There are three possible cases, Case 1. τ 2n ≤τ ∧τ 2n+1 . Then y 1;; 2n ∧ 2n ∧ 2n+1 =y 1; 2n+1 ; 2n ∧ 2n ∧ 2n+1 =U 1 2n (3.39) and two BSDEs share the same parameters, so y 1;; 2n t =y 1; 2n+1 ; 2n t . 34 Case2. τ 2n+1 ≤τ∧τ 2n . Usingtheargument(3.29),(3.35),wegetτ 2n+1 = ˜ τ 2n+1 . Since Y is the snell envelope of y, we have y 1; 2n+1 ; 2n 2n+1 =Y 1; 2n ˜ 2n+1 ≥y 1;; 2n 2n+1 (3.40) Case 3. τ ≤τ 2n ∧τ 2n+1 . If τ 2n+1 ≤τ 2n , the argument is similar to Case 2. y 1; 2n+1 ; 2n =Y 1; 2n ≥y 1;; 2n (3.41) If τ 2n+1 ≥τ 2n , Y 1; 2n and y 1; 2n+1 ; 2n share the same dynamics, hence equal to each other. In fact, by the definition of τ 2n+1 in (3.29), ˜ τ 2n+1 = τ 2n , hence K 2n = K t using Theorem 2.9. As the result y 1; 2n+1 ; 2n =Y 1; 2n ≥y 1;; 2n Similarly we can prove J 2 (τ 2n+1 ,τ)≤J 2 (τ 2n+1 ,τ 2n+2 ). Now define τ ∗ 1 △ = lim n→∞ τ 2n+1 and τ ∗ 2 △ = lim n→∞ τ 2n . (3.42) We shall prove that (τ ∗ 1 ,τ ∗ 2 ) is a NEP. We divide the proof into several lemmas. Lemma 3.12. Assume Assumption 2.1, 2.2 and 2.3. (i) For any τ ∈T 0 , we have lim n→∞ J 1 (τ,τ 2n ) =J 1 (τ,τ ∗ 2 ). (ii) For any τ ∈T 0 , we have lim n→∞ J 2 (τ 2n+1 ,τ) =J 2 (τ ∗ 1 ,τ). 35 proof. Apply BSDE stability Theorem 2.5 directly to y 1;; 2 and y 1;; 2n . we need to check all the conditions. |ξ n −ξ| = |(U 1 2n −U 1 2 )1 {≥ 2n } +(U 1 2 −L 1 )1 { 2 ≤< 2n } | ≤ |U 1 2n −U 1 2 |1 {≥ 2n } +|U 1 2 −L 1 |1 { 2 ≤< 2n } (3.43) By Cauchy-Schwartz inequality, lim n→∞ E|ξ n −ξ| 2 ≤ 2·(lim n→∞ E|U 1 2n −U 1 2 | 2 +lim n→∞ E{|U 1 2 −L 1 |1 { 2 ≤< 2n } } 2 ) ≤ 2·(lim n→∞ E|U 1 2n −U 1 2 | 2 +C 2 ·lim n→∞ P(τ ∗ 2 ≤τ <τ 2n )) (3.44) U is bounded and continuous, so E|U 1 2n −U 1 2 | 2 ≤C 2 and lim n→∞ U 1 2n =U 1 2 . By the bounded convergence theorem, the first term of (3.44) vanishes. Since lim n→∞ τ 2n =τ ∗ 2 the second term vanishes. f n −f = (1 {< 2n } −1 {< 2 } )f 1 (t,0,0) = 1 { 2 ≤< 2n } f 1 (t,0,0) (3.45) so lim n→∞ E{ ∫ T 0 |f n −f| 2 } = lim n→∞ E{ ∫ T 0 1 { 2 ≤< 2n } f 2 1 (t,0,0)dt} ≤ Clim n→∞ P(τ ∗ 2 ≤τ <τ 2n ) (3.46) which vanishes. 36 Combine (3.44) and (3.46), and notice that f n (t,y,z) → f(t,y,z), P −a.s. We could apply Theorem 2.5 now, which implies that lim n→∞ J 1 (τ,τ 2n ) =J 1 (τ,τ ∗ 2 ). Similarly, we could show (ii). The proof is complete. Lemma 3.13. Assume Assumption 2.1, 2.2 and 2.3. Then it holds that lim n→∞ J 1 (τ 2n+1 ,τ 2n ) =J 1 (τ ∗ 1 ,τ ∗ 2 ); lim n→∞ J 2 (τ 2n−1 ,τ 2n ) =J 2 (τ ∗ 1 ,τ ∗ 2 ). Proof. WestillapplyTheorem2.5andshowthestabilityofthesystem. Denote ξ n = L 1 2n+1 1 { 2n+1 < 2n } +U 1 2n 1 { 2n+1 ≥ 2n } ξ = L 1 1 1 { 1 < 2 } +U 1 2 1 { 1 ≥ 2 } |ξ n −ξ| = |L 1 2n+1 1 { 2n+1 < 2n } −L 1 1 1 { 1 < 2 } +U 1 2n 1 { 2n+1 ≥ 2n } −U 1 2 1 { 1 ≥ 2 } | ≤ |L 1 2n+1 1 { 2n+1 < 2n } −L 1 1 1 { 2n+1 < 2n } +L 1 1 1 { 2n+1 < 2n } −L 1 1 1 { 1 < 2 } | +|U 1 2n 1 { 2n+1 ≥ 2n } −U 1 2 1 { 2n+1 ≥ 2n } +U 1 2 1 { 2n+1 ≥ 2n } −U 1 2 1 { 1 ≥ 2 } | (3.47) 37 By Cauchy-Schwartz inequality, E|ξ n −ξ| 2 ≤ 4·(E|L 1 2n+1 1 { 2n+1 < 2n } −L 1 1 1 { 2n+1 < 2n } | 2 +E|L 1 1 1 { 2n+1 < 2n } −L 1 1 1 { 1 < 2 } | 2 +E|U 1 2n 1 { 2n+1 ≥ 2n } −U 1 2 1 { 2n+1 ≥ 2n } | 2 +E|U 1 2 1 { 2n+1 ≥ 2n } −U 1 2 1 { 1 ≥ 2 } | 2 ) (3.48) It is easy to check every term vanishes as n →∞, like what we did in (3.44). Also the other conditions in stability satisfy, so using the stability conclusion in Theorem 2.5 we finish (i). Similarly we could prove (ii). We are now ready to show that (τ ∗ 1 ,τ ∗ 2 ) is a NEP. Theorem 3.14. Under Assumption 2.1, 2.2 and 2.3, the non-zero-sum Dynkin game associated with J 1 and J 2 in (3.25) has a NEP (τ ∗ 1 ,τ ∗ 2 ). proof. First, by Lemma 3.12 (i), Lemma 3.13, and Lemma 3.4, we have J 1 (τ,τ ∗ 2 )≤J 1 (τ ∗ 1 ,τ ∗ 2 ), ∀τ ∈T 0 . (3.49) Similarly, for any τ such that P(τ =τ ∗ 1 <T) = 0, we have J 2 (τ ∗ 1 ,τ)≤J 2 (τ ∗ 1 ,τ ∗ 2 ). (3.50) 38 In the general case, for any τ ∈T 0 , set ˆ τ n △ =        [τ + 1 n ]∧T, if τ =τ ∗ 1 <T; τ, otherwsie. Then ˆ τ n is a stopping time and P(ˆ τ n =τ ∗ 1 <T) = 0. Thus (3.50) leads to J 2 (τ ∗ 1 ,ˆ τ n )≤J 2 (τ ∗ 1 ,τ ∗ 2 ). Send n→∞, we obtain (3.50) for general τ. Combine (3.49) and (3.50), we know (τ ∗ 1 ,τ ∗ 2 ) is a NEP. Remark3.15. ItisstraightforwardtocombineresultsTheorem3.2and3.14,that isincorporatingrecursiveutilityintonon-zero-sumDynkingamewithmulti-player. 39 Chapter 4 Dynamic Non-Zero-Sum Dynkin Game Problem and System of RBSDEs 4.1 Introduction Previous chapter discussed the existence of non-zero-sum Dynkin game in static setting, i.e. fixing initial time at 0, and find an equilibrium. In static setting, players choose their own strategies at initial time 0 only , hope everything goes well and fall asleep until one of their strategies is actually exercised. Cvitani´ c and Karatzas [11] constructed the bridge between RBSDEs and zero-sum Dynkin Games. Natural questions are what value functions dynamics of non-zero-sum Dynkin game are, and whether there are parallel results to Cvitani´ c and Karatzas via BSDE and RBSDE. Inreality,playersmayregretabouttheirinitialcommitmentandwanttoadjust basedonnewinformation. Dynamicproblemisconcernedwiththisissue,wewould 40 like to discuss how the dynamics of value function Y 1 andY 2 are evolving through time,moreimportantlywhetherplayersstickwiththeirinitialstrategies. Ourmain result of Chapter 4 is that value dynamics Y 1 and Y 2 of a non-zero-sum Dynkin game satisfy a RBSDE system with jumping terms under a class of equilibrium called time-autonomous. First, we want to motivate readers by a simple example showing that the static Dynkin game problem may conceive multiple NEP, leading to different utilities. Here is an example illustrated by a figure. 41 x x x x x x x x x x t 1 L 1 1 L 2 1 t 2 U 2 2 L 2 2 U 1 2 L 1 2 t 3 L 1 3 U 2 3 t 4 U 2 L 2 U 1 L 1 Figure 4.1: An Example of Equilibrium Nonuniqueness 42 Example4.1. Hamad` eneandZhang[28]constructedanNEP,whiletheirmethod discriminated neither players, nor the stopping strategies τ (or σ) with same util- ities. Let us use deterministic utility process L i and U i , i = 1,2 to illustrate this point. (i) If we start from player 1 and enforce τ 0 be t 3 , one NEP would be (τ ∗ ,σ ∗ ) = (t 3 ,t 2 ) and (Y 1 ,Y 2 ) = (U 1 t 2 ,L 2 t 2 ). (ii) If we start from player 2 and make σ 0 be t 4 = T, then τ 1 = t 1 , σ 1 = t 4 . Another NEP would be (τ ∗ ,σ ∗ ) = (t 1 ,t 4 ) and (Y 1 ,Y 2 ) = (L 1 t 1 ,U 2 t 1 ). It shows under the non-zero-sum frame work, both NEP and utilities may not be unique. To study value dynamics of non-zero-sum Dynkin game, we have to be carefulaboutwhichequilibriumpointfamilyandwhatplayer’svaluesareselected. Toovercomethedifficultyofvaluefunctionnon-uniqueness,wewanttodisentangle one player’s strategy from the other one as much as possible. The idea is inspired by construction of NEP via RBSDE [11]. Denition 4.2. Under Assumption 2.1, 2.2 and 2.3, we have showed that static non-zero-sum Dynkin game has NEP for any time t ∈ [0,T] and denote a family of NEP as {(τ t ,σ t )} t∈[0;T] ∈{T t ×T t }. We say that NEP {(τ t ,σ t )} t∈[0;T] is time- autonomous if :        τ s =τ t , ∀ t≤s≤τ t σ s =σ t , ∀ t≤s≤σ t (4.1) 43 4.2 Some Preparation Results Remark 4.3. If we want to define stopping strategies in Dynkin game, some properties about NEP τ t and σ t must be satisfied first. For example, Y 1 t =L 1 t , Y 2 t =U 2 t and Y 2 t =L 2 t , Y 1 t =U 1 t Remark 4.4. With definition of time-autonomous NEP{(τ t ,σ t )} 0≤t≤T , we notice that value dynamics Y 1 and Y 2 should be supermartingales locally at [t,σ t ] and [t,τ t ] respectively. Bear this in mind, a RBSDE system is discovered as the follow- ing.                                                                dY 1 t =Z 1 t dB t −dK 1 t Y 1 t ≥L 1 t K 1 ∈S ci 1 {Y 2 t >L 2 t or Y 1 t ̸=U 1 t } dK 1 t = 0 dY 2 t =Z 2 t dB t −dK 2 t Y 2 t ≥L 2 t K 2 ∈S ci 1 {Y 1 t >L 1 t or Y 2 t ̸=U 2 t } dK 2 t = 0 (4.2) 44 A non-zero-sum Dynkin game is defined as below: Each player a i is associated with two payoff/cost processes L i , U i . Their expected utilities J i (t,τ,σ), i = 1,2, are defined as follows: J 1 (t,τ,σ) △ = E t { L 1 1 {≤} +U 1 1 {<} } J 2 (t,τ,σ) △ = E t { L 2 1 {<} +U 2 1 {≤} } . (4.3) Theorem 4.5. Assume L,U ∈ S 2 . Suppose RBSDE system (4.2) has a solution (Y 1 ,Y 2 ). Define stopping times τ t △ = inf { s≥t :Y 1 s =L 1 s ,Y 2 s =U 2 s } σ t △ = inf { s≥t :Y 2 s =L 2 s ,Y 1 s =U 1 s } ∀t≤T (4.4) then,{(τ t ,σ t )} 0≤t≤T is a time-autonomous NEP of the corresponding non-zero-sum Dynkin game (4.3) and Y 1 ,Y 2 are value functions of the game, i.e. Y 1 t △ =J 1 (t,τ t ,σ t ) =E t { L 1 t 1 {t≤t} +U 1 t 1 {t<t} } Y 2 t △ =J 2 (t,τ t ,σ t ) =E t { U 2 t 1 {t≤t} +L 2 t 1 {t<t} } (4.5) proof. We need to show that ∀t, (i) fixσ t , the stopping timeτ t is the optimal stopping time, andY 1 t is the value function. (ii) fix τ t , the stopping time σ t is the optimal stopping time, and Y 2 t is the value function, i.e. 45 Y 1 t = J 1 (t,τ t ,σ t ) =E t { L 1 t 1 {t≤t} +U 1 t 1 {t<t} } = esssup ≥t J 1 (t,τ,σ t ) = esssup ≥t E t { L 1 1 {≤t} +U 1 t 1 {t<} } Y 2 t = J 2 (t,τ t ,σ t ) =E t { U 2 t 1 {t≤t} +L 2 t 1 {t<t} } = esssup ≥t J 2 (t,τ t ,σ) = esssup ≥t E t { U 2 t 1 {t≤} +L 2 1 {<t} } (4.6) (i) From the definition ofτ t andσ t , we know thatτ t ̸=σ t . Fixσ t , the dynamics of Y 1 in (4.2) becomes                        Y 1 t =U 1 t − ∫ t t Z 1 r dB r +K 1 t −K 1 t Y 1 t ≥L 1 t K 1 ∈S ci ∫ t t 1 (Y 1 r >L 1 r or Y 2 r ̸=U 2 r ) dK 1 r = 0 (4.7) Weclaimthatthesolutionof(4.7)isexactlythesameasasolutionofaregular RBSDE:                        y 1 t =U 1 t − ∫ t t z 1 r dB r +k 1 t −k 1 t y 1 t ≥L 1 t k 1 ∈S ci ∫ t t (y 1 t −L 1 t )dk 1 t = 0 (4.8) 46 Based on the classical RBSDE Theorem 2.9: RBSDE (4.8) exists a unique solution which is the smallest solution among all satisfying (i) and (ii) in (4.8) as well. Y satisfies (i) and (ii) in (4.8), hence Y 1 ≥y 1 . Next, if RBSDE (4.7) has a solution, then it satisfies RBSDE                        Y 1 t =U 1 t − ∫ t t Z 1 r dB r +K 1 t −K 1 t Y 1 t ≥L 1 t −|Y 2 t −U 2 t | K 1 ∈S ci ∫ t t (Y 1 r −L 1 r +|Y 2 r −U 2 r |)dK 1 r = 0 (4.9) Based on the RBSDE comparison Theorem 2.8: L 1 t ≥ L 1 t −|Y 2 t −U 2 t | implies y 1 ≥Y 1 . Hence Y 1 =y 1 in [t,σ t ]. Then, we show the first half of (4.6). In fact, ∀τ ≥t, Y 1 t = Y 1 ∧t − ∫ ∧t t Z 1 r dB r +K 1 ∧t −K 1 t ≥ Y 1 ∧t − ∫ ∧t t Z 1 r dB r , since K is increasing = E t { Y 1 ∧t } = E t { Y 1 1 {<t} +Y 1 t 1 {t<} } ≥ E t { L 1 1 {<t} +Y 1 t 1 {t<} } , since Y ≥L (4.10) 47 as well as, Y 1 t = Y 1 t∧t − ∫ t∧t t Z 1 r dB r +K 1 t∧t −K 1 t = Y 1 t∧t − ∫ t∧t t Z 1 r dB r = E t { Y 1 t∧t } = E t { L 1 t 1 {t<t} +U 1 t 1 {t<t} } (4.11) Hence, τ t is optimal stopping time in (4.6). (ii) Similarly, we could show the second half of (4.6). Theorem 4.6. Assume L,U ∈ S 2 . Suppose non-zero-sum Dynkin game (4.3) has a time-autonomous NEP {(τ t ,σ t )} 0≤t≤T , and Y 1 ,Y 2 are value functions of the game, i.e. Y 1 t △ = J 1 (t,τ t ,σ t ) =E t { L 1 t 1 {t≤t} +U 1 t 1 {t<t} } = esssup ≥t J 1 (t,τ,σ t ) = esssup ≥t E t { L 1 1 {≤t} +U 1 t 1 {t<} } Y 2 t △ = J 2 (t,τ t ,σ t ) =E t { U 2 t 1 {t≤t} +L 2 t 1 {t<t} } = esssup ≥t J 2 (t,τ t ,σ) = esssup ≥t E t { U 2 t 1 {t≤} +L 2 1 {<t} } (4.12) In addition, if both U 1 and U 2 are supermartingales. Then Y 1 satisfies a RBSDE system (4.2) locally on [t,σ t ], and Y 2 satisfies a RBSDE system (4.2) locally on [t,τ t ]. proof. we need to check (4.2) one by one. 48 (i) we claim that Y 1 is a supermartingale. ∀s≥t, E t { Y 1 s } = E t { Y 1 s 1 {t<s} +Y 1 s 1 {t≥s} } = E t { E s [L 1 s 1 {s≤s} +U 1 s 1 {s<s} ]1 {t<s} +Y 1 s 1 {t≥s} } ≤ E t { E s [U 1 s 1 {s≤s} +U 1 s 1 {s<s} ]1 {t<s} +Y 1 s 1 {t≥s} } = E t { E s [U 1 s∧s ]1 {t<s} +Y 1 s 1 {t≥s} } ≤ E t { U 1 s 1 {t<s} +Y 1 s 1 {t≥s} } since U is a supermartingale = E t { E t [U 1 s 1 {t<s} ]+Y 1 s 1 {t≥s} } = E t { E t [U 1 s ]1 {t<s} +Y 1 s 1 {t≥s} } ≤ E t { U 1 t 1 {t<s} +Y 1 s 1 {t≥s} } = E t { Y 1 t 1 {t<s} +Y 1 s 1 {t≥s} } = E t { Y 1 t∧s } ≤ Y 1 t since Y is a supermartingale before σ t (4.13) (ii) Y 1 t △ = esssup ≥t J 1 (t,τ,σ t ) = esssup ≥t E t { L 1 1 {≤t} +U 1 t 1 {t<} } ≥ E t { L 1 t 1 {t≤t} +U 1 t 1 {t<t} } ≥ L 1 t (4.14) 49 (iii) and (iv) on t 0 ∈ {Y 2 t > L 2 t or Y 1 t ̸= U 1 t }, t 0 < σ t 0 . Since σ t 0 is a time- autonomous NEP, then ∀t∈ [t 0 ,σ t 0 ], σ t =σ t 0 . Hence, Y 1 t = esssup ≥t E t { L 1 1 {≤t 0 } +U 1 t 0 1 {t 0 <} } (4.15) Apply RBSDE and Optimal stopping time properties Theorem 2.9, before σ t 0 , Y 1 t is the solution of a RBSDE                Y 1 t =U 1 t 0 − ∫ t 0 t Z 1 r dB r +K 1 t 0 −K 1 t Y 1 t ≥L 1 t ∫ t 0 t (Y 1 r −L 1 r )dK 1 r = 0 (4.16) where K 1 ∈S ci . Similarly, we could Y 2 t satisfies the dynamics in (4.2). Remark 4.7. We have to indicate in Theorem 4.6, dynamics of Y 1 andY 2 locally satisfy (4.2). When one player, say Player 2 reaches her stopping strategy, i.e. t = σ t , value utility of Player 1 Y 1 has a natural tendency to jump right after t =σ t . Vice versa. If we want to find a generalized dynamics globally, we have to involve jump term. The observation is the initiative of the next section. 50 4.3 Decomposition and Equivalency On non-zero-sum Dynkin game, there is no connection between (L 1 ,L 2 ) and (U 2 ,U 1 ). We have to decompose original continuous process {K + t } (or {K − t }) [26] into two processes: (1) K 1 t : K 1 ∈S ci to Y 1 t =L 1 t , where L∈S 2 . (2) N 1 t : N 1 is a jump process related to Y 2 t =L 2 t . By this decomposition, we could see the structure more clearly. We propose a new RBSDE system with jumps,                                                                Y 1 t =Y 1 T − ∫ T t Z 1 s dB s +(K 1 T −K 1 t )−(N 1 T −N 1 t ) Y 1 t ≥L 1 t {(Y 1 t −L 1 t )+|Y 2 t −U 2 t |}dK 1 t = 0, K 1 ∈S ci {(Y 2 t −L 2 t )+|Y 1 t −U 1 t |}dN 2 t = 0 Y 2 t =Y 2 T − ∫ T t Z 2 s dB s +(K 2 T −K 2 t )−(N 2 T −N 2 t ) Y 2 t ≥L 2 t {(Y 2 t −L 2 t )+|Y 1 t −U 1 t |}dK 2 t = 0, K 2 ∈S ci {(Y 1 t −L 1 t )+|Y 2 t −U 2 t |}dN 2 t = 0 (4.17) Next theorem is the main result of this section which demonstrates the equiv- alence between RBSDE system and Dynkin game. 51 Theorem 4.8. (a) Suppose Dynkin game has a time-autonomous NEP {(τ t ,σ t )} 0≤t≤T , then denote value function (Y 1 ,Y 2 ) as Y 1 t △ =J 1 (t,τ t ,σ t ) =E t { L 1 t 1 {t≤t} +U 1 t 1 {t<t} } Y 2 t △ =J 2 (t,τ t ,σ t ) =E t { U 2 t 1 {t≤t} +L 2 t 1 {t<t} } (4.18) Then the dynamics of Y 1 and Y 2 satisfy a RBSDE system (4.17). (b) Suppose RBSDE system (4.17) has a solution (Y i ,Z i ,K i ,N i ), i = 1,2. Define stopping times τ t △ = inf { s≥t :Y 1 s =L 1 s ,Y 2 s =U 2 s } σ t △ = inf { s≥t :Y 2 s =L 2 s ,Y 1 s =U 1 s } (4.19) Then{(τ t ,σ t )} 0≤t≤T is a time-autonomous NEP of the corresponding non-zero- sum Dynkin game (4.3) and Y 1 ,Y 2 are value functions of the game. proof of part (a). We need to show that Y 1 and Y 2 defined by (4.18) satisfy (4.17). Define D △ = {t : t < σ t }. It is easily to see that D is a right open set, i.e. ∀t∈D, ∃ϵ> 0 s.t. t+ϵ∈D. On the other end, Y 1 t =U 1 t when t∈D c (1) When t∈ [s,σ s ), by definition time-autonomous NEP (4.2) σ t =σ s . Hence Y 1 t is a supermartingale in [s,σ s ) and dY 1 t =Z 1 t dB t −dK 1 t by Theorem 2.9 52 (2) Consider σ 0 WLOG, Y 1 t =J 1 (t,τ t ,σ 0 ) where 0≤t≤σ 0 . Y 1 t = J 1 (t,τ t ,σ 0 ) = esssup ≥t J 1 (t,τ,σ 0 ) ≥ E t { L 1 t 1 {t< 0 } +U 1 0 1 { 0 ≤t} } = L 1 t (4.20) (3) When t ∈ D and {Y 1 t > L 1 t or Y 2 t ̸= U 2 t }, then t < τ t ∧σ t . By definition of time-autonomous NEP, Y 1 t is a martingale. Hence K 1 t = 0 on {Y 1 t > L 1 t or Y 2 t ̸= U 2 t }. When t∈D c which means Y 1 t = U 1 t , Y 2 t = L 1 t , all jumps are included in term N 1 . (4) As suggested in (1), Y 1 is a supermartingale in D which means N 1 = 0. In D c , Y 1 t = U 1 t , Y 2 t = L 1 t . We have already checked each item for Y 1 so far. Similarly, we could check them for Y 2 as well. Next, we briefly described how to construct the solution of RBSDE (4.17) with time-autonomous NEP {(τ t ,σ t )} 0≤t≤T in (a). ∀t ∈ D c , let Y 1 t = U 1 t ,L 2 t = L 2 t . When t∈D, we construct solution on [t,σ t ) by solving                Y 1 t =U 1 t + ∫ t t Z 1 s dB s −dK 1 t Y 1 t ≥L 1 t (Y 1 t −L 1 t )dK 1 t = 0, K 1 ∈S ci (4.21) 53 After solving each interval of [t,σ t ), let Z 1 t △ = Z 1 t 1 (t∈D) and K 1 t △ = K 1 t 1 (t∈D) . With Y,Z,K in hand, let dN 1 t △ =dY 1 t +dZ 1 t −dK 1 t . We could construct (Y 2 ,Z 2 ,K 2 ,N 2 ) similarly. proof of part (b). We need to show that ∀t∈ [0,T], (i) fix σ t , τ t is the optimal stopping time, and Y 1 t is the value function, (ii) fix τ t , σ t is the optimal stopping time, and Y 2 t is the value function, i.e. Y 1 t = J 1 (t,τ t ,σ t ) =E t { L 1 t 1 {t≤t} +U 1 t 1 {t<t} } = esssup ≥t J 1 (t,τ,σ t ) = esssup ≥t E t { L 1 1 {≤t} +U 1 t 1 {t<} } Y 2 t = J 2 (t,τ t ,σ t ) =E t { U 2 t 1 {t≤t} +L 2 t 1 {t<t} } = esssup ≥t J 2 (t,τ t ,σ) = esssup ≥t E t { U 2 t 1 {t≤} +L 2 1 {<t} } (4.22) (i) From the definition of τ t andσ t , we know thatτ t ̸=σ t . Fixσ t , the dynamics of Y 1 in (4.17) becomes                        Y 1 t =U 1 t − ∫ t t Z 1 r dB r +K 1 t −K 1 t Y 1 t ≥L 1 t K 1 ∈S ci ∫ t t 1 (Y 1 r >L 1 r or Y 2 r ̸=U 2 r ) dK 1 r = 0 (4.23) 54 We claim that the solution of (4.23) is exactly the same as a solution of a regular RBSDE:                        y 1 t =U 1 t − ∫ t t z 1 r dB r +k 1 t −k 1 t y 1 t ≥L 1 t k 1 ∈S ci ∫ t t (y 1 t −L 1 t )dk 1 t = 0 (4.24) ApplyRBSDEcomparisonTheorem2.8,RBSDE(4.24)existsauniquesolution which is the smallest solution among all satisfying (i) and (ii) in (4.24) as well. Y satisfies (i) and (ii) in (4.23), hence Y 1 ≥y 1 . Next, if RBSDE (4.23) has a solution, then it satisfies RBSDE                        Y 1 t =U 1 t − ∫ t t Z 1 r dB r +K 1 t −K 1 t Y 1 t ≥L 1 t −|Y 2 t −U 2 t | K 1 ∈S ci ∫ t t (Y 1 r −L 1 r +|Y 2 r −U 2 r |)dK 1 r = 0 (4.25) Apply RBSDE comparison Theorem 2.8, L 1 t ≥L 1 t −|Y 2 t −U 2 t |, y 1 ≥Y 1 . Hence Y 1 =y 1 in [t,σ t ]. 55 Then, we show the first half of (b). In fact, ∀τ ≥t, Y 1 t = Y 1 ∧t − ∫ ∧t t Z 1 r dB r +K 1 ∧t −K 1 t ≥ Y 1 ∧t − ∫ ∧t t Z 1 r dB r , K 1 ∈S ci = E t { Y 1 ∧t } = E t { Y 1 1 {<t} +Y 1 t 1 {t<} } ≥ E t { L 1 1 {<t} +Y 1 t 1 {t<} } , since Y ≥L (4.26) as well as, Y 1 t = Y 1 t∧t − ∫ t∧t t Z 1 r dB r +K 1 t∧t −K 1 t = Y 1 t∧t − ∫ t∧t t Z 1 r dB r = E t { Y 1 t∧t } = E t { L 1 t 1 {t<t} +U 1 t 1 {t<t} } (4.27) Hence, τ t is optimal stopping time. (ii) Similarly, we could show the second half of (b). Remark4.9. Li[40],andKaratzasandLi[31,32]studiednon-zero-sumstochastic differential games of control and stopping. They introduced a system of RBSDEs for the optimal stopping problem. However, their RBSDE system is somewhat differentfromoursandtheydidn’testablishtheequivalencybetweentheirRBSDE system and the original game problem. 56 4.4 Three Special Solutions to the Equivalent RBSDE System We have already set up the connection between RBSDE system (4.17) and Dynkin Game (4.3). In this section we would like to briefly discuss three special solutions to (4.17). We find that none of these cases could cover the rest. One of the future research possibilities is to unify these cases. (1) If L 1 = −U 2 and L 2 = −U 1 , i.e. zero-sum Dynkin game, then RBSDE system (4.17) has unique solution. The proof was due to Cvitani´ c and Karatzas [11], improved by Hamad` ene and Hasanni [26]. More generally, assume M =L 1 +U 2 =L 2 +U 1 , where M is a martingale, the whole theory remains the same. (2) If L 1 (or L 2 ) is a supermartingale, then Player 1 (or Player 2) has incentive to exercise the contract immediately. There is a trivial solution Y 1 = L 1 , Y 2 =U 2 aswellastheoptimalstoppingtimeτ t =t,σ t =T. (orrespectively, Y 1 =U 1 , Y 2 =L 2 as well as the optimal stopping time τ t =T,σ t =t.) (3) If U 1 s >L 1 t ,∀t,s∈ [0,T] (or U 2 s >L 2 t ,∀t,s∈ [0,T]), then Player 1 (or Player 2) has incentive to stay in the contract as long as possible. There is another NEP where τ t = T,σ t determined using Snell Envelope optimal stopping Theorem 2.9 and 2.10. (or respectively, σ t = T,τ t determined using Snell 57 Envelope optimal stopping Theorem 2.9 and 2.10.) In this case, Y 1 t = U 1 t , Y 2 =L 2 t (or Y 1 t =L 1 t , Y 2 =U 2 t ) 4.5 Discrete Time Stopping Framework In this section, we assume player 1 and 2 only stop at a discrete set Θ of time. In a more rigorous setting, Assumption 4.10. ∀ω ∈ Ω, τ(ω) ∈ Θ △ = {0,1,··· ,n}. Since τ is a stopping time, ∀t∈ [0,T], there must have {τ ≥t} ={τ ≥⌈t⌉}∈F ⌈t⌉ We denote all such stopping times on discrete set Θ as T Θ . Assume τ,σ∈T Θ . Weexploretheexistenceanduniquenessoftime-autonomousNEP(TANEPfor short) under discrete time framework. One existence theorem and one uniqueness claim are presented in the following. Theorem 4.11. Assume Assumption 4.10, then TANEP always exists. proof. We use induction to show this theorem. When |Θ| = 1, τ = σ = T = 0 is the only choice. Suppose the conclusion in theorem is true for |Θ| =n. 58 For |Θ| = n+1, i.e. Θ = {0,1,··· ,n}. By induction assumption, ∃ TANEP {(ˆ τ t ,ˆ σ t )} t=1;2;···;n on Θ\{0}. Consider the whole time horizon now. There are three cases in the following (i) J 1 (0,0,ˆ σ 1 )≥J 1 (0,ˆ τ 1 ,ˆ σ 1 ) ; (ii) J 1 (0,0,ˆ σ 1 )<J 1 (0,ˆ τ 1 ,ˆ σ 1 ), while J 2 (0,ˆ τ 1 ,0)≥J 2 (0,ˆ τ 1 ,ˆ σ 1 ) ; (iii) J 1 (0,0,ˆ σ 1 )<J 1 (0,ˆ τ 1 ,ˆ σ 1 ), and J 2 (0,ˆ τ 1 ,0)<J 2 (0,ˆ τ 1 ,ˆ σ 1 ). Case (i) Let        τ 0 = 0 σ 0 = ˆ σ 1 (τ i ,σ i ) = (ˆ τ i ,ˆ σ i ) i = 1,2,··· ,n. Then {(τ i ,σ i )} i∈Θ is a TANEP. Case(ii) Let        τ 0 = ˆ τ 1 σ 0 = 0 (τ i ,σ i ) = (ˆ τ i ,ˆ σ i ) i = 1,2,··· ,n. Then {(τ i ,σ i )} i∈Θ is a TANEP. Case(iii) Let        τ 0 = ˆ τ 1 σ 0 = ˆ σ 1 (τ i ,σ i ) = (ˆ τ i ,ˆ σ i ) i = 1,2,··· ,n. Then {(τ i ,σ i )} i∈Θ is a TANEP. 59 Combine all these three cases, we showed∃ TANEP on a finite number of time Θ ={0,1,··· ,n}. Thus, ∃ TANEP under Assumption 4.10. Next,weconsideruniquenessofthevaluefunctionJ 1 andJ 2 underAssumption 4.10. Claim 4.12. There are two different sources causing non-uniqueness of TANEP, even value functions. (a) Player 1 and Player 2 have incentive to stop at the same time. Specifically,        J 1 (0,0,ˆ σ 1 )≥J 1 (0,ˆ τ 1 ,ˆ σ 1 ) J 2 (0,ˆ τ 1 ,0)≥J 2 (0,ˆ τ 1 ,ˆ σ 1 ) (4.28) A tentative condition E s [L 1 t +L 2 t + ˆ U 1 t ∧ ˆ U 2 t ]>L 1 s +L 2 s ∀s<t (4.29) could rule it out, where ˆ U △ =U −L. (b) Player 1 (or 2) has options to stop immediately or some time later to get the same amount of utility. Specifically, J 1 (0,0,ˆ σ 1 ) =J 1 (0,ˆ τ 1 ,ˆ σ 1 ) (4.30) (orJ 2 (0,ˆ τ 1 ,0) =J 2 (0,ˆ τ 1 ,ˆ σ 1 )) (4.31) 60 argument. First, we consider (a) in our proof only, assume (b) never happens. We still use induction to show uniqueness of TANEP. When |Θ| = 1, it is obvious value function J 1 ,J 2 are unique. Suppose the value function J 1 ,J 2 are unique for |Θ| =n. When |Θ| =n+1 Consider the whole time horizon now. There are following three cases (i) J 1 (0,0,ˆ σ 1 )≥J 1 (0,ˆ τ 1 ,ˆ σ 1 ) ; (ii) J 1 (0,0,ˆ σ 1 )<J 1 (0,ˆ τ 1 ,ˆ σ 1 ), while J 2 (0,ˆ τ 1 ,0)≥J 2 (0,ˆ τ 1 ,ˆ σ 1 ) ; (iii) J 1 (0,0,ˆ σ 1 )<J 1 (0,ˆ τ 1 ,ˆ σ 1 ), and J 2 (0,ˆ τ 1 ,0)<J 2 (0,ˆ τ 1 ,ˆ σ 1 ) . Case(iii) It is true in this case, since no one would stop at 0 bearing a lower utility. In fact, player 1 and 2 would not stop at the same time other than time T. WLOG, player 2 would stop after time 1, so it will stop at ˆ σ 1 . Since J 1 (0,0,ˆ σ 1 )< J 1 (0,ˆ τ 1 ,ˆ σ 1 ), player 1 would stop at ˆ τ 1 . Since J 1 (0,ˆ τ 1 ,ˆ σ 1 ) =E[J 1 (1,ˆ τ 1 ,ˆ σ 1 )] J 2 (0,ˆ τ 1 ,ˆ σ 1 ) =E[J 2 (1,ˆ τ 1 ,ˆ σ 1 )] TANEP is unique. Case(ii) If J 1 (0,0,ˆ σ 1 )<J 1 (0,ˆ τ 1 ,ˆ σ 1 ) and J 2 (0,ˆ τ 1 ,0)>J 2 (0,ˆ τ 1 ,ˆ σ 1 ) 61 Player 1 would not stop at 0, otherwise as the argument in (i), player 2 would stop at ˆ σ 1 , which contradicts J 1 (0,0,ˆ σ 1 )<J 1 (0,ˆ τ 1 ,ˆ σ 1 ). So player 1 would stop at ˆ τ 1 , and J 2 (0,ˆ τ 1 ,0)>J 2 (0,ˆ τ 1 ,ˆ σ 1 ) implies σ 0 = 0. Thus, J 1 (0) =U 1 0 ,J 2 (0) =L 2 0 . Similarly,wecouldshowinthesymmetriccase,thevaluefunctionwithTANEP is still unique. Case(i) If J 1 (0,0,ˆ σ 1 )>J 1 (0,ˆ τ 1 ,ˆ σ 1 ) and J 2 (0,ˆ τ 1 ,0)>J 2 (0,ˆ τ 1 ,ˆ σ 1 ) There are two possible TANEP, one is (τ 0 ,σ 0 ) = (0,ˆ σ 1 ), (τ i ,σ i ) = (ˆ τ i ,ˆ σ i ) i = 1,2,··· ,n, which leads to J 1 (0) = L 1 ,J 2 (0) = U 2 ; the other is (τ 0 ,σ 0 ) = (ˆ τ 1 ,0), (τ i ,σ i ) = (ˆ τ i ,ˆ σ i ) i = 1,2,··· ,n, which leads to J 1 (0) = U 1 ,J 2 (0) = L 1 . Appar- ently, these two possible TANEP leads to different value functions (J 1 (0),J 2 (0)). If we want to stick to the uniqueness of TANEP even value functions, we have to rule out this situation. The condition J 1 (0,0,ˆ σ 1 )>J 1 (0,ˆ τ 1 ,ˆ σ 1 ) and J 2 (0,ˆ τ 1 ,0)>J 2 (0,ˆ τ 1 ,ˆ σ 1 ) 62 is equivalent to L 1 0 >E[L 1 1 ∧ 1 + ˆ U 1 1 ∧ 1 1 { 1 < 1 } ] L 2 0 >E[L 2 1 ∧ 1 + ˆ U 2 1 ∧ 1 1 { 1 < 1 } ] Hence, L 1 0 +L 2 0 > E[L 1 1 ∧ 1 +L 2 1 ∧ 1 + ˆ U 1 1 ∧ 1 1 { 1 < 1 } + ˆ U 2 1 ∧ 1 1 { 1 < 1 } ] ≥ E[L 1 1 ∧ 1 +L 2 1 ∧ 1 + ˆ U 1 1 ∧ 1 ∧ ˆ U 2 1 ∧ 1 1 { 1 < 1 } + ˆ U 1 1 ∧ 1 ∧ ˆ U 2 1 ∧ 1 1 { 1 < 1 } ] = E[L 1 1 ∧ 1 +L 2 1 ∧ 1 + ˆ U 1 1 ∧ 1 ∧ ˆ U 2 1 ∧ 1 ] This is immediate contradiction to rule out condition (4.29) E s [L 1 t +L 2 t + ˆ U 1 t ∧ ˆ U 2 t ]>L 1 s +L 2 s ∀s<t Hence, both value functions and TANEP are unique under Assumption 4.10 and condition (4.29). Now let us briefly discuss situation (b). If J 1 (0,0,ˆ σ 1 ) = J 1 (0,ˆ τ 1 ,ˆ σ 1 ), player 1 has two options, either stops at 0 or stays until ˆ τ 1 . He will get L 1 0 in either case. 63 To guarantee the uniqueness of value function, we have to enforce J 2 (0,0,ˆ σ 1 ) = J 2 (0,ˆ τ 1 ,ˆ σ 1 ) i.e. L 1 0 =E[L 1 1 1 { 1 ≤ 1 } +U 1 1 1 { 1 < 1 } ] U 2 0 =E[L 2 1 1 { 1 < 1 } +U 2 1 1 { 1 ≤ 1 } ] A possible way to make it true is that linking L 1 with U 2 ; L 2 with U 1 in a flavor of zero-sum Dynkin game. For example, define        L 2 =−c·U 1 +M U 2 =−c·L 1 +M where c is a positive constant, M is a martingale. Remark4.13. IndiscretesettingwithAssumption4.10,itmaybepossibletopick L and U elaborately to avoid (4.30). When considering continuous time Dynkin game, in general (4.30) is naturally embedding in the structure. 4.6 Semimartingale Framework We want to extend our results to continuous semimartingale framework: both L 1 and L 2 are semimartingales, while Assumption 2.2 preserves. 64 Theorem 4.14. Consider a finite time horizon [0,T]. Assume both L 1 and L 2 are either submartingales or supermartingales in interval [i,i+1],∀i∈N. Also L 1 and L 2 are supermartingales at the same interval. Then TANEP always exist. proof. ByequivalencyTheorem4.8, aslongasweconstructasolutiontoequiv- alent system (4.17), there exist TANEP. We use induction to construct solution to RBSDE system (4.17). When n = 1, (i) if L 1 and L 2 are submartingales, let K 1 =N 1 =K 2 =N 2 ≡ 0 Y 1 t = L 1 1 − ∫ 1 t Z 1 s dB s Y 2 t = U 2 1 − ∫ 1 t Z 2 s dB s which implies Y 1 t = E t [L 1 1 ]≥L 1 t Y 2 t = E t [U 2 1 ]≥L 2 t so (Y,Z,K,N) is well defined, with (τ t ,σ t ) = (1,1) the optimal strategy. (ii) if L 1 is supermartingale, L 2 is submartingale. Let N 1 =K 2 =Z 2 ≡ 0 Y 1 t = L 1 1 − ∫ 1 t Z 1 s dB s +K 1 T −K 1 t Y 2 t = U 2 1 −(N 1 T −N 1 t ) 65 which implies Y 1 t = L 1 t dL 1 t = Z 1 t dB t −dK 1 t Y 2 t = U 2 t dN 2 t = dU 2 t with (τ t ,σ t ) = (t,1). (iii) similar to (ii), L 2 is supermartingale, L 1 is submartingale. Define ˜ τ △ = inf{s≥ 0 :L 2 s =E s [U 2 1 ]}. ∀t∈ [0,˜ τ], let (τ t ,σ t ) = (1,t). Let N 2 =K 1 =Z 1 ≡ 0 Y 1 t = U 1 t dN 1 t = dU 1 t Y 2 t = L 2 t dY 2 t = Z 2 t dB t −dK 2 t ∀t∈ [˜ τ,1], let (τ t ,σ t ) = (1,1), K 1 =N 1 =K 2 =N 2 ≡ 0 Y 1 T = L 1 T dY 1 t = Z 1 t dB t Y 2 T = U 2 T dY 2 t = Z 2 t dB t 66 Which finish the construction. By comparison Theorem 2.8, with identical initial value, we know that Y 2 t ≥L 2 t . Suppose for [1,n] the conclusion is true. For [0,n], by assumption, there exists TANEP on [1,n]. We want to complete TANEP on [0,1] as well. The following proof is almost the same as previous argument. (i) L 1 and L 2 are submartingales. Let K 1 =N 1 =K 2 =N 2 ≡ 0 Y 1 t = Y 1 1 − ∫ 1 t Z 1 s dB s Y 2 t = Y 2 1 − ∫ 1 t Z 2 s dB s where Y 1 1 =J 1 (1,τ 1 ,σ 1 ), Y 2 1 =J 2 (1,τ 1 ,σ 1 ). Y 1 t = E t [Y 1 1 ]≥L 1 t Y 2 t = E t [Y 2 1 ]≥L 2 t so (Y,Z,K,N) is well defined, with (τ t ,σ t ) = (1,1) the optimal strategy. 67 (ii) Define ˜ τ △ = inf{s≥ 0 : L 1 s = E s [L 1 1 1 { 1 ≤ 1 } +U 1 1 1 { 1 < 1 } ]}. ∀t∈ [0,˜ τ], let (τ t ,σ t ) = (t,σ 1 ). Let N 1 =K 2 =Z 2 ≡ 0 Y 1 t = L 1 t dL 1 t = Z 1 t dB t −dK 1 t Y 2 t = U 2 t dN 2 t = dU 2 t everything is well defined. ∀t ∈ [˜ τ,1], let (τ t ,σ t ) = (τ 1 ,σ 1 ), K 1 = N 1 = K 2 = N 2 ≡ 0 Y 1 t = Y 1 1 − ∫ 1 t Z 1 s dB s Y 2 t = Y 2 1 − ∫ 1 t Z 2 s dB s which implies Y 1 t = E t [Y 1 1 ] and Y 2 t = E t [Y 2 1 ] which finish the construction. By definition of ˜ τ, Y 1 ˜ = L 1 ˜ . By comparison Theorem 2.8, with identical initial value we know that Y 1 t ≥L 1 t on [˜ τ,1]. (iii) is similar to (ii). Hence, we finish the proof by induction. In the above theorem, we rule out the case that both L 1 and L 2 are super- martingales. Next theorem shows that if this is the case, TANEP may not exist. Claim 4.15. If L 1 and L 2 are supermartingales in some interval before maturity, TANEP may not exist. 68 argument. WLOG, suppose ∀t ∈ [1,n], ∃ TANEP, and L 1 and L 2 are super- martingales in [0,1]. Define ˜ τ △ ={s≥ 0 :τ s+ =τ 1 }∧1 ˜ σ △ ={s≥ 0 :σ s+ =σ 1 }∧1 We claim that L 1 ˜ = E ˜ [L 1 1 1 { 1 ≤ˆ } +U 1 ˆ 1 {ˆ < 1 } ] = sup ≥˜ E ˜ [L 1 1 {≤ˆ } +U 1 ˆ 1 {ˆ < 1 } ] where ˆ σ △ = σ ˜ In fact, left hand side is always larger by NEP. By continuity of L 1 , if left hand side is strictly larger, player 1 could still stop immediately after ˜ τ which contradicts to the definition of ˜ τ. Since player 1 and player 2 could not stop at the same time, we could always find ξ s.t. ξ < ˜ τ and ∀t ∈ (ξ,˜ τ], σ t = ˆ σ. i.e. we could extend ˜ τ to a left local interval, player 2 would not stop at this interval. One hand, ∀t∈ (ξ,˜ τ] L 1 t ≥ E t [L 1 ˜ ] = E t [sup ≥˜ E ˜ [L 1 1 {≤ˆ } +U 1 ˆ 1 {ˆ < 1 } ]] ≥ sup ≥˜ E t [E ˜ [L 1 1 {≤ˆ } +U 1 ˆ 1 {ˆ < 1 } ]] = sup ≥˜ E t [L 1 1 {≤ˆ } +U 1 ˆ 1 {ˆ <} ] 69 On the other hand, L 1 t ≥ sup t≤≤˜ E t [L 1 ] = sup t≤≤˜ E t [L 1 1 {≤ˆ } +U 1 ˆ 1 {ˆ <} ] combine above two inequalities, we get L 1 t ≥ sup t≤ E t [L 1 1 {≤ˆ } +U 1 ˆ 1 {ˆ <} ] so τ t = t on (ξ,˜ τ]. Especially, τ + = ξ+. By continuity of L and U as well as L < U, we see that σ = ˆ σ, i.e. we could actually extend to a closed interval [ξ,˜ τ]. Using the above argument, we could extend again. Finally, such extendable interval is [0,˜ τ], i.e. Y 1 t =L 1 t ,Y 2 t =U 2 t on [0,˜ τ]. Similarly, we could get another extendable interval [0,˜ σ], i.e. Y 2 t = L 2 t ,Y 1 t = U 1 t . Apparently, as long as the intersection of above two extendable intervals, the system is incompatible. 70 Chapter 5 Principle-Agent Problem on Stopping Times 5.1 Introduction In Chapter 2 and Chapter 3, we studied stochastic Dynkin game. Players involved in Dynkin game are symmetrical, in the sense that no one has more bargaining power than the others. The players are indifferent about switching their index, and the switching of the position won’t hurt or benefit any of them. One may connect Dynkin game with Cournot game [10] and Bertrand game [5]. But this assumption is not always the case as what really happens on the market. In many situationsmarketplayersmayhavedifferentresourcesandinformationswhichlead to different perspectives and levels of power. This phenomenon motivates us to consider another type of stopping time problem, Principal-Agent problem. In recent years there has been a significant increase of interest in continuous- timePrincipal-Agentproblems,orcontracttheory,andtheirapplications. Cvitani´ c and Zhang did extensive study on continuous-time Principal-Agent problems, the 71 interested readers may find [12] a good reference book on this topic. Our research hereisthefirstwhichtoattackPrincipal-Agentproblemspurelyonstoppingtimes. One may connect the Principal-Agent problem with Stackelberg leadership model [23]. Stackelberg leadership model is a strategic game in economics in which the leader firm moves first and then the follower firm moves sequentially. Theproblemissolvedbytwostepoptimizations. SolvingPrincipal-Agentproblem has the same flavor where value is unique, while a time-inconsistent issue is raised. The time-inconsistent formally means the Bellman’s equation and dynamical pro- gramming principal do not apply here, see example 5.4. 5.2 Formulation of the Problem To motivate the problem, consider a temporary employment contract in a labor market. Company (principal) may firstly propose a termination date of a tempo- rary contract to individual contractor base on the project scale, economic environ- ment, its own financial condition and the most important, contractor’s reaction. Thecontractor(agent)thenhavetherighttomanagehis/herownquittingtimeor simply stay in the contract until the termination as the reaction of the proposal. With the anticipative knowledge of contractor’s reaction, any rational company would maximize its own benefit by proposing a well selected termination date at the beginning round. 72 Assume we have a system controlled by principal and agent a 1 and a 2 . The system works or is alive up to the time when one of the them decides to stop the control at a stopping time τ for a 1 and σ for a 2 . When the system is stopped, the payment for a 1 (resp. a 2 ) amounts to a quantity J 1 (τ,σ) (resp. J 2 (τ,σ)) which could be negative and then it is a cost. To simplify the problem, we define principle’s and agent’s utility as follows. J 1 (τ,σ) =E[U ∧ ] (5.1) J 2 (τ,σ) =E[L ∧ ] (5.2) where L, U ∈ D . Notice that there is no comparison between process U and L through the paper from now on. Denition 5.1. We say that the Principal-Agent problem with J 1 and J 2 has a solution (an optimal point) if there exists a pair of stopping times (τ ∗ ,σ(τ ∗ )) such that the Agent’s problem is σ(τ)∈{σ :J 2 (τ,σ) = sup J 2 (τ,ξ)}, ∀τ ∈T (5.3) and the Principle’s problem is to maximize his own utility by choosing τ ∗ , i.e. J 1 (τ ∗ ,σ(τ ∗ )) = sup J 1 (τ,σ(τ)) (5.4) 73 Notice that by Snell envelope Theorem 2.10, for any Principle’s strategy τ, the Agent’s optimal stopping time exists and denote it as σ(τ), provided some mild condition, L∈D. Remark 5.2. When J 1 +J 2 ≡ 0, (τ ∗ ,σ(τ ∗ )) exists and it satisfies J 1 (τ,σ(τ ∗ ))≤J 1 (τ ∗ ,σ(τ ∗ ))≤J 1 (τ ∗ ,σ), for any τ, σ. We call such a (τ ∗ ,σ(τ ∗ )) a saddle-point for the problem. Additionally this exis- tence implies in particular that: inf 2 sup 1 J 1 (τ 1 ,τ 2 ) = sup 1 inf 2 J 1 (τ 1 ,τ 2 ), i.e. values of Principal-Agent problem and Dynkin game are essentially the same in this particular case. Remark 5.3. WLOG, we could always assume that σ≤τ, since we could always enforce σ equal to τ if any σ larger than τ. This assumption does not affect the value of J 1 and J 2 . We identify a time-inconsistent issue in Principal-Agent problem even in deter- ministic case. Here is the example illustrated by a figure. 74 t =t 0 t 1 L ˜ (s) U ˜ (s) t 2 s =t 3 L (s) U (s) t 4 t 5 L (t) U (t) t 6 U L t 6 Figure 5.1: An Example of Time Inconsistency 75 Example 5.4. We construct a deterministic example to see clearly what we call time-inconsistent issue. Both L and U are finitely bounded and moving upward and downward a limited times. At time t=t 0 : Consider the static problem. If player 1 chooses his stopping strategy τ t ≥ t 5 , say τ t = t 6 , then player 2 would select his stopping strategy at σ(τ t ) =t 5 , since this rule brings his maximum benefit among all points in L. If player 1 chooses other stopping time, the choice left to player 2 is before t 5 . In any case, player 1 is even or worse off. We are convinced that both player 1 and 2 are waiting until t 5 . (τ t ,σ(τ t )) = (t 6 ,t 5 ) is an optimal strategy when t = 0. At time s=t 3 : Consider the static Principal-Agent problem again. Player 1 wants to stop immediately at t 3 , and it leaves player 2 no room but stopping at t 3 . (τ s ,σ(τ s )) = (t 3 ,t 3 ) is an optimal strategy when s =t 3 . Remark 5.5. The first observation is when time elapses from t tos, principal and agent’s stopping strategy change from (t 6 ,t 5 ) to (t 3 ,t 3 ), and the initial stopping strategy (t 6 ,t 5 ) is never implemented. Another observation is E t [Y 1 s ] =U t 3 >U t 5 =Y 1 t (5.5) These observations mean principal and agent may change their minds about orig- inal stopping time choices. In the extreme case, principal and agent are busy of choosing new strategies, while regretting about their decisions they just made. 76 Players in Dynkin game do not regret about their strategies, rather they adjust their strategies when one player’s strategy is actually implemented . We could see this point by construction of NEP [11] or [28]. We call this phenomena time- inconsistent. In Principal-Agent problem, it is difficult to study, even define value dynamics with time-inconsistent fact. RecentdevelopmentonmathematicaltreatmentofAssetAllocationandBehav- ioral Finance recalled time-inconsistent issue, see e.g. [7, 8, 24, 30, 37, 42, 51, 56, 57, 58] and references therein. Bj¨ ork and Mergoci [7] provided a general the- ory of Markovian time-inconsistent stochastic control problems. They indicated two basic ways of handling time inconsistency. One is pre-committed approach where investors choose optimal strategy at time zero and stick with their origi- nal commitments. The other approach is to study time-inconsistent issue within game theoretic framework. Game theoretic approach essentially modify the target functions of investors, applying to our study, transfer Principal-Agent problem to Dynkin game. What we use in this paper is the first approach, that is to study the static Principal-Agent problem since it is already interesting and difficult enough. We propose a method to solve discrete Principal-Agent problem using Lagrange Multipliers. In the following section, we assume Principle and Agent stop at time of a dis- crete set Θ ={0,1,2,··· ,n}. We want to design the mechanism how to construct both Principle’s strategy τ ∗ and Agent’s strategy σ ∗ to this P-A problem. We 77 analyze the solution to P-A problem when n=2 and 3 to help understanding first, then introduce the lagrange multipliers method to multi-period P-A problem. 5.3 2-Period Problem Let Θ ={0,1,2}. When τ ≡ 0 or σ≡ 0, naively J 1 =U 0 and J 2 =L 0 . Otherwise, we assume both τ and σ are greater than 1. We could rewrite stopping times as sum of indicator functions i.e. τ = 1+1 A , where A∈F 1 σ = 1+1 B , where B ∈F 1 (5.6) more precisely, we have ∀ω ∈ A, τ(ω) = 2; ∀ω / ∈ A, τ(ω) = 1. σ is similar. First, given A we want to solve Agent problem: J 2 (τ,σ(τ)) = sup J 2 (τ,σ) = sup B∈F 1 E[L 1 1 B c +L 2 1 B ] , where B ⊆A = sup B∈F 1 E[L 1 +(L 2 −L 1 )1 B ] = sup B∈F 1 E[L 1 +L 2 1 1 B ] (5.7) where L 2 1 △ =E 1 [L 2 ]−L 1 . 78 To maximize the expression (5.7), we need to choose B(A) s.t. {{L 2 1 > 0}∩A}⊆B(A)⊆{{L 2 1 ≥ 0}∩A} . Second, we want to solve Principle problem: J 1 (τ ∗ ,σ(τ ∗ )) = sup J 1 (τ,σ(τ)) = sup A∈F 1 E[U 1 1 B(A) c +U 2 1 B(A) ] (5.8) Similarly, we define U 2 1 △ =E 1 [U 2 ]−U 1 and rewrite the equation as, sup A∈F 1 E[U 1 1 B(A) c +U 2 1 B(A) ] = sup A∈F 1 E[U 1 +U 2 1 1 B(A) ] = sup A∈F 1 E[U 1 +U 2 1 1 A∩{L 2 1 >0} ] (5.9) To maximize the expression (5.8), we need to choose A ∗ s.t. {{L 2 1 ≥ 0}∩{U 2 1 > 0}}⊆A ∗ ⊆{{L 2 1 ≥ 0}∩{U 2 1 ≥ 0}} . Final step would be comparing J 1 (τ ∗ ) where τ ∗ ≥ 1 with J 1 (0) = U 0 . So far, we have solved 2-period P-A problem so far. 79 5.4 3-Period Problem Let Θ = {0,1,2,3}. When τ ≡ 0 or σ ≡ 0, naively J 1 = U 0 and J 2 = L 0 . Otherwise, let τ = 1+1 A 1 +1 A 1 1 A 2 , where A 1 ∈F 1 A 2 ∈F 2 σ = 1+1 B 1 +1 B 1 1 B 2 , where B 1 ∈F 1 B 2 ∈F 2 (5.10) We also assume σ≤τ i.e. B 1 ⊆A 1 and B 2 ⊆A 2 . First, given A 1 and A 2 which means with the knowledge of τ, we want to solve Agent problem: J 2 (τ,σ(τ)) = sup J 2 (τ,σ) = sup B 1 ;B 2 E[L 1 1 B c 1 +L 2 1 B 1 B c 2 +L 3 1 B 1 B 2 ] = sup B 1 ;B 2 E[L 1 +(L 2 −L 1 )1 B 1 +(L 3 −L 2 )1 B 1 1 B 2 ] = sup B 1 ;B 2 E[L 1 +L 2 1 1 B 1 +L 3 2 1 B 1 1 B 2 ] = sup B 1 ;B 2 E[L 1 +1 B 1 (L 2 1 +1 B 2 L 3 2 )] (5.11) where L j i △ =E i [L j ]−L i and U j i △ =E i [U j ]−U i , ∀i≤j. We need to choose A 2 ∩ {L 3 2 > 0} ⊆ B 2 (A 2 ) ⊆ A 2 ∩ {L 3 2 ≥ 0} A 1 ∩ {L 2 1 +E 1 (1 A 2 L 3+ 2 )> 0} ⊆ B 1 (A 1 ,A 2 ) ⊆ A 1 ∩ {L 2 1 +E 1 (1 A 2 L 3+ 2 )≥ 0} (5.12) to maximize Agent’s utility. 80 Second, solve Principle problem: J 1 (τ ∗ ,σ(τ ∗ )) = sup J 1 (τ,σ(τ)) = sup A 1 ;A 2 E[U 1 +1 B 1 (A 1 ;A 2 ) (U 2 1 +1 B 2 (A 2 ) U 3 2 )] = sup A 1 ;A 2 E { U 1 +1 A 1 1 {L 2 1 +E 1 (1 A 2 L 3+ 2 )≥0} [U 2 1 +E 1 [1 A 2 1 {L 3 2 ≥0} U 3 2 ] } (5.13) Condition on period 1, we only need to maximize the following term: sup A 2 1 {L 2 1 +E 1 (1 A 2 L 3+ 2 )≥0} [U 2 1 +E 1 [1 A 2 1 {L 3 2 ≥0} U 3 2 ] or rewrite into a simpler form, Problem 5.6. sup A E(1 A Y) s.t. c 2 +E(1 A X)≥ 0 where X =L 3+ 2 Y =U 3 2 c 2 =L 2 1 (5.14) 81 In fact, X ≥ 0, so if we enlarge A, c 2 +E(1 A X) is always increasing. Hence we always put {Y ≥ 0} in A. sup A E(1 A Y) = sup A E(Y1 Y≥0 )+E(Y1 A ) where A − △ ={A\{Y ≥ 0}}⊆{Y < 0} s.t. c 2 +E(X1 Y≥0 )+E(X1 A )> 0 i.e. inf A E(Y − 1 A ) s.t. c 3 △ = (c 2 +E(X1 Y≥0 )) − ≤E(X1 A ) (5.15) (i)if c 3 = 0, we take A − =∅ (ii)if c 3 > 0, define Z △ = Y X in {Y < 0}∩{X > 0} and Ω z ={Z ≤z}. Lemma 5.7. If there exists z 0 s.t. E(1 Ωz 0 X) = c 3 . Then A − could take the form of Ω z 0 . 82 proof. ∀A − s.t. E(X1 A )≥c 3 , E(Y − 1 A )−E(Y − 1 Ωz 0 ) = E(Y − 1 A \Ωz 0 )−E(Y − 1 Ωz 0 \A ) = E(XZ1 A \Ωz 0 )−E(XZ1 Ω z 0 \A ) ≥ z 0 [E(X1 A \Ωz 0 )−E(X1 Ωz 0 \A )] = z 0 [E(X1 A )−E(X1 Ωz 0 )] ≥ 0 (5.16) AftergettingoptimalA ∗ 2 usingLemma5.7, wedetermineA ∗ 1 in(5.13). Optimal A ∗ 1 ={U 2 1 +E 1 [1 A 2 1 {L 3 2 ≥0} U 3 2 ]≥ 0} based on the selectedA ∗ 2 . τ ∗ > 0 is determined byA ∗ 1 andA ∗ 2 . Finally, we compare J 1 (τ ∗ ) where τ ∗ > 0 with J 1 (0) = U 0 to determine the best τ ∗ . So far, we have solved the 3-period Principle problem. 5.5 Multi-Period Problem Inspired by solving 2-period and 3-period problems, we want to extend our argu- ment to multi-period Principal-Agent problem in this section. Formal arguments 83 are provided in the following, while notations are already getting very heavy. We hopetoaddresstheproblemaswellascontinuouscaserigorouslyinfutureresearch. Suppose Θ ={0,1,··· ,n}. When τ or σ is 0, J 1 =U 0 and J 2 =L 0 . If both τ and σ is greater than 1, denote τ = 1+ ∑ n−1 m=1 ∏ m i=1 1 A i , where A i ∈F i σ = 1+ ∑ n−1 m=1 ∏ m i=1 1 B i , where B i ∈F i (5.17) We also assume σ≤τ i.e. B i ⊆A i , ∀i = 1,2,··· ,n. Firstly, Agent’s problem could be written as, J 2 (τ,σ(τ)) = sup J 2 (τ,σ) = sup B i ; i=1;···;n−1 E{L 1 + ∑ n−1 m=1 [L m+1 m ∏ m i=1 1 B i ]} (5.18) 84 where L j i △ = E i [L j ]−L i and U j i △ = E i [U j ]−U i . We assume Principal has already selected his/her strategy (A), to optimize (5.18) B n−1 (A n−1 ) = A n−1 ∩ {L n n−1 ≥ 0} B n−2 (A n−2 ,A n−1 ) = A n−2 ∩ {L n−1 n−2 +E n−2 (1 A n1 L n+ n−1 )≥ 0} B n−3 (A n−3 ,A n−2 ,A n−1 ) = A n−3 ∩ {L n−2 n−3 +E n−3 [ 1 A n2 (L n−1 n−2 +E n−2 (1 A n1 L n+ n−1 )) + ]> 0} . . . . . . B 1 (A 1 ,··· ,A n−1 ) = A 1 ∩ {L 2 1 +E 1 [1 A 2 (L 3 2 + E 2 (1 A 3 (L 4 3 +E 3 (1 A 4 ···)) + )) + ]> 0} (5.19) Secondly, Principal’s problem is as follows, J 1 (τ ∗ ) = sup J 1 (τ,σ(τ)) = sup A i ; i=1;···;n−1 E{U 1 + ∑ n−1 m=1 [U m+1 m ∏ m i=1 1 B i (A i ;···;A n1 ) ]} (5.20) Claim 5.8. We claim that A must take the following form A n−1 = {−U n n−1 ≤z n−2 L n n−1 }∩{L n n−1 ≥ 0} A n−2 = {−g n−2 (z n−2 )≤z n−1 f n−2 (z n−2 )}∩{f n−2 (z n−2 )≥ 0} . . . . . . A 1 = {−g 1 (z 1 ,··· ,z n−1 )≤z 0 f 1 (z 1 ,··· ,z n−1 )}∩{f 1 (z 1 ,··· ,z n−1 )≥ 0} (5.21) 85 where, f n−1 △ = U n n−1 g n−1 △ = L n n−1 f n−2 (x) △ = L n−1 n−2 +E n−2 [1 {−g n1 ≤xf n1 }∩{f n1 ≥0} f n−1 ] g n−2 (x) △ = U n−1 n−2 +E n−2 [1 {−g n1 ≤xf n1 }∩{f n1 ≥0} g n−1 ] f n−3 (x,z n−2 ) △ = L n−2 n−3 +E n−3 [1 {−g n2 (z n2 )≤xf n2 (z n2 )}∩{f n2 (z n2 )≥0} f n−2 (z n−2 )] g n−3 (x,z n−2 ) △ = U n−2 n−3 +E n−3 [1 {−g n2 (z n2 )≤xf n2 (z n2 )}∩{f n2 (z n2 )≥0} g n−2 (z n−2 )] . . . . . . (5.22) and z 0 ,z 1 ,··· ,z n−2 to be determined. With Claim 5.8, we introduce Lagrange Multipliers method to solve constraint optimization problem. Consider the following problems. Problem 5.9. A constrained optimization problem. Define I 0 △ = inf Z≥z 0 E[1 (h(Z)≤z) g(Z)] s.t. E[1 (h(Z)≤z) f(Z)] =c (5.23) where f,g are increasing functions, z,c are constants and h(x) △ = f(x) g(x) . We want to find Z over all random variables which are larger than a constant z 0 . 86 Problem5.10. Asimilarconstrainedoptimizationproblemisdefinedasfollowing. Define I 1 △ = inf Z≥z 0 E[1 (h(Z)≤z) g(Z)] s.t. E[1 (h(Z)≤z) f(Z)]≥c (5.24) where the notations are the same as Problem (5.9). Problem 5.11. An unconstrained optimization problem. Define I(λ) △ = inf Z≥z 0 E[1 (h(Z)≤z) g(Z)+1 (h(Z)≤z) λf(Z)]−λc Z △ = arginf Z≥z 0 E[1 (h(Z)≤z) g(Z)+1 (h(Z)≤z) λf(Z)]−λc λ ∗ ∈ {λ :E[1 (h(Z )≤z) f(Z )] =c} (5.25) Lemma 5.12. I(λ ∗ ) =I 0 If we restrict the value λ ∗ ≤ 0, then we also have, I(λ ∗ ) =I 1 proof. By the definition of I 0 , since E[1 (h(Z )≤z) f(Z )] =c, we have I(λ ∗ )≥I 0 87 On the other hand, ∀Z s.t. E[1 (h(Z)≤z) f(Z)] =c E[1 (h(Z)≤z) g(Z)] =E[1 (h(Z)≤z) g(Z)+1 (h(Z)≤z) λf(Z)]−λc By the definition of I(λ), we have I(λ ∗ )≤I 0 I(λ ∗ )≥I 1 is similar to the previous proof. If λ ∗ ≤ 0, ∀Z s.t. E[1 (h(Z)≤z) f(Z)]≥c E[1 (h(Z)≤z) g(Z)]≥E[1 (h(Z)≤z) g(Z)+λ ∗ (1 (h(Z)≤z) f(Z)−c)]≥I(λ ∗ ) Hence, we proved I(λ ∗ ) =I 1 . From Lemma 5.12, we transform a constraint problem to an unconstrained problem by Lagrange Multipliers method. To further simplify the notation, define G(ω,Z(ω),λ) = g(ω,Z(ω)) + λf(ω,Z(ω)). The unconstrained Problem 5.11 becomes ∀z,λ, inf Z≥z 0 E[1 (h(Z)≤z) G(Z,λ)] (5.26) and the solution is Z (ω) = argmin x≥z 0 1 (h(!;x)≤z) G(ω,x,λ) = argmin x∈h 1 (!;(−∞;z])∩[z 0 ;∞) G(ω,x,λ) (5.27) 88 Problem 5.13. A constrained optimization problem, inf z 2 ;z 3 E 1 [1 (h 2 (z 2 ;z 3 )≤z 1 ) g 2 (z 2 ,z 3 )] s.t. E 1 [1 (h 2 (z 2 ;z 3 )≤z 1 ) f 2 (z 2 ,z 3 )]≥c 1 (ω) ω∈F 1 and E 2 [1 (h 3 (z 3 )≤z 1 ) f 3 (z 3 )]≥c 2 (ω) ω∈F 2 (5.28) could be transformed to another unconstrained optimization problem, Problem 5.14. inf z 2 ;z 3 E 1 [1 (h 2 (z 2 ;z 3 )≤z 1 ) g 2 (z 2 ,z 3 ) +λ 1 1 (h 2 (z 2 ;z 3 )≤z 1 ) (f 2 (z 2 ,z 3 )−c 1 ) +λ 2 1 (h 3 (z 3 )≤z 2 ) (f 3 (z 3 )−c 2 )] (5.29) We define the solution to problem (5.14) as z 1 ; 2 2 and z 1 ; 2 3 . Then over all possible λ 1 ,λ 2 , find λ ∗ 1 ,λ ∗ 2 s.t. constrained becomes equality, i.e. E 1 [1 (h 2 (z 1 ; 2 2 ;z 1 ; 2 3 )≤z 1 ) f 2 (z 1 ; 2 2 ,z 1 ; 2 3 )] = c 1 (ω) ω∈F 1 E 2 [1 (h 3 (z 1 ; 2 3 )≤z 2 ) f 3 (z 1 ; 2 3 )] = c 2 (ω) ω∈F 2 (5.30) For multi-period constraint optimization problem, the procedure is similar to what we described above. 89 Bibliography [1] Alariot, M., Lepeltier, J. P., and Marchal, B. (1982): Jeux de Dynkin. 2nd Bad Honnef Workshop on Stochastic Processes, Lecture Notes in Control and Inform. Sci., Springer-Verlag, Berlin, 23-32. [2] Alvarez, H.R.Luis (2008): A Class of Solvable Stopping Games. Appl. Math. Optim., DOI 10.1007/s00245-008-9035-z [3] Bensoussan, A. and Friedman, A. (1977): Nonzero-Sum Stochastic Differen- tial Games with Stopping Times and Free Boundary Value Problem. Trans. A.M.S. 213 (2), 275-327. [4] Bensoussan, A. and Friedman, A. (1974): Non-Linear Variational Inequalities and Differential Games with Stopping Times. J. Funct. Anal. 16, 305-352. [5] Bertrand, J. (1883): Th´ eorie math´ ematique de la richesse sociale. Journal des Savants 499-508. [6] Bismut, J.M. (1977): Sur un probl` eme de Dynkin. Z. Wahrsch. Verw. Geb., 39, 31-53. [7] Bj¨ ork, T. and Mergoci, A. (2010): A General Theory of Markovian Time Inconsistent Stochastic Control Problems. [8] Bj¨ ork, T., Mergoci, A. and Zhou, X. (2011): Mean-Variance Portfolio Opti- mization with State Dependent Risk Aversion. [9] Cattiaux, P. and Lepeltiet, J.P. (1990): Existence of an Quasi-Markov Nash Equilibrium for Nonzero Sum Markov Stopping Games. Stochastics and stochastics reports, 30 (2), 85-103. 90 [10] Cournot, A. (11838): Recherches sur les Principles Mathematiques de la The- orie des Rochesses. English edition: Researches into the Mathematical Prin- ciples of the Theory of Wealth, ed. N. Bacon Macmillan, 1897. [11] Cvitanic, J. and Karatzas, I. (1996): Backward SDEs with Reflection and Dynkin Games. Annals of Probability, 24 (4), 2024-2056. [12] Cvitanic,J.andZhang,J.(2012): ContractTheoryinContinuous-TimeMod- els. Springer Finance. [13] Duffie, D. and Epstein, L. (1992): Stochastic Differential Utility, Economet- rica, 60, 353-394. [14] Duffie, D. and Epstein, L. (1992): Asset Pricing with Stochastic Differential Utilities , Review of Financial Studies, 5, 411-436. [15] Dellacherie, C. and Meyer, P. A. (1980): Probabilit´ es et Potentiel, Chapitres V-VIII. Hermann, Paris. [16] Dynkin, E.B. (1969): A Game-Theoretic Version of an Optimal Stopping Problem. Soviet Math. Dokl. 10, 270-274. [17] Ekstrom, E., Peskir, G. (2008): Optimal Stopping Games fo Markov Pro- cesses. SIAM JCO, 47 (2), 684-702. [18] Ekstrom,E.,Villeneuve,S.(2006): OntheValueofOptimalStoppingGames. Annals of Applied Probability, 16 (3), 1576-1596. [19] ElKaroui,N.(1980): Lesaspectsprobabilistesducontrˆ olestochastique.Ecole d’´ et´ e de probabilit´ es de Saint-Flour, Lect. Notes in Math. No 876, Springer Verlag. [20] El Karoui, N., Peng, S. and Quenez, M. C. (1997): Backward Stochastic Dif- ferential Equations in Finance Mathematical Finance, January 1997, Volume 7, Issue 1, Pages 1C71. [21] ElKaroui,N.,Kapoudian,C.,Pardoux,E.,Peng,S.andQuenez,M.C.(1997): Reflected Soulutions of Backward SDE’s, and Related Obstacle Problems for PDE’s. The Annals of Problability, Vol. 25, No. 2, 702-737 [22] Etourneau, E. (1986): R´ esolution d’un probl` eme de jeu de somme non nulle sur les temps d’arrˆ et. Th` ese de 3-i` eme cycle, Univ. Paris 6. [23] Fudenberg, D. and Tirole, J(1991): Game Theory. The MIT Press 12-13, 67-69 91 [24] Grenadier, S.R. and Wang, N.: Investment under Uncertainty and Time- Inconsistent Preferences. preprint. [25] Hamad` ene, S.(2006): MixedZero-SumDiffrentialGameandAmericanGame Options. SIAM J. Control Optim. 45, 496-518. [26] Hamad` ene, S. and Hassani, M. (2005): BSDEs with Two Reflecting Barriers: the General Result. Probability Theory Relatd Fields 132, 237-264. [27] Hamad` ene,S.andHassani,M.(2011): TheMulti-playerNonzero-sumDynkin Game in Continuous Time. http://arxiv.org/pdf/1110.5889.pdf. [28] Hamad` ene, S. and Zhang, J. (2009): The Continuous Time Nonzero-Sum Dynkin Game Problem and Application in Game Options. [29] Harsanyi, J. (1967): Games with Incomplete Information Played by ”Bayesian” Players, I-III. Part I. The Basic Model. Management Science, Vol. 14, No. 3, Theory Series. [30] Herings, P.J. and Rohde, K.I.M.: Time-Inconsistent Preferences in a General Equilibrium Model. preprint. [31] Karatzas, I. and Li, Q. (2009): Martingale Interpretation to a Non-Zero-Sum Stochastic Differential Game of Controls and Stoppings. preprint. [32] Karatzas, I. and Li, Q. (2011): A BSDE Approach to Non-Zero-Sum Stochas- tic Differential Games of Controls and Stoppings. preprint. [33] Karatzas, I. and Shreve, S. (1998): Methods of Mathematical Finance Appli- cations of Mathematics: stochastic Modelling and Applied Probability 39. Springer [34] Karatzas, I.andSudderth, W.(2005): StochasticGamesofControlandStop- ping for a Linear Diffusion. [35] Karatzas, I. and Zamfirescu, I. (2008): Martingale Approach to Stochastic Differential Games of Control and Stopping The Annals of Probability, 2008, Vol. 36, No. 4, 1495-1527. [36] Kifer, Y. (2000): Game Options. Finance and Stochastics, 4, 443-463. [37] Krusell, P. and Smith , A. A., Jr. (2003): Consumption and Saving Decisions with Quasi-Geometric Discounting. Econometrica, 71 (2003), 366-375. [38] Laraki, R. and Solan, E. (2005): The Value of Zero-Sum Stopping Games in Continuous Time. SIAM J. Control Optim., 43, 1913-1922. 92 [39] Lepeltier, J. P. and Maingueneau, M.A. (1984): Le Jeu de Dynkin en Th´ eorie G´ en´ erale Sans l’Hypoth` ese de Mokobodski. Stochastics, 13, 25-44. [40] Li, Q. (2009): Two Approaches to Non-Zero-Sum Stochastic Differential Games of Controls and Stoppings. PH.D. Thesis. [41] Markowitz,H.(1952): PortfolioSelection.JournalofFinance7(1952), 77-98. [42] Miller, M. and Salmon, M. (1985): Dynamic Games and the Time Incon- sistency of Optimal Policy in Open Economics. The Economic Journal, 95 (1985), 124-137. [43] Morimoto, H. (1984): Dynkin Games and Martingale Methods. Stochastics, 13, 213-228. [44] Morimoto, H. (1986): Nonzero-Sum Discrete Parameter Stochastic Games with Stopping Times. Probab. Th. Rel. Fields, 72, 155-160. [45] Myerson,R.B.(1991): GameTheory: AnalysisofConflict.HarvardUniversity Press, p. 1. Chapter-preview links, pp. vii-xi. [46] Nagai, H. (1987): Nonzero-Sum Stopping Games of Symmetric Markov Pro- cesses. Probability Theory and Related Fields ´ Eiteur, 75 (4), 487-497. [47] Nash, J.F. (1950): Equilibrium Points in N-person Games. Proceedings of the National Academy of Sciences 36 (36): 48C9, DOI:10.1073/pnas.36.1.48, PMC 1063129, PMID 16588946, MR0031701. [48] Neumann, J. v. (1928). Zur Theorie der Gesellschaftsspiele Mathematische Annalen, 100(1), p p. 295-320. English translation: On the Theory of Games of Strategy, in A. W. Tucker and R. D. Luce, ed. (1959). Contributions to the Theory of Games, v. 4, p p. 13-42. [49] Neveu,J.(1975).Discrete-ParameterMartingales.English translation: North- Holland, Amesterdam and American Elsevier, New York. [50] Ohtsubo, Y. (1987): A Nonzero-Sum Extension of Dynkin Stopping Problem. Mathematics of Operations Research, 12 (2), 277-296. [51] Palacios-Huerta, I.(2003): Time-InconsistentPreferencesinAdamSmithand Davis Hume. History of Political Economy, 35 (2003), 241-268. [52] Selten, R. (1965): Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetr¨ agheit: Teil I: Bestimmung des dynamischen Preisgleichgewichts. Zeitschrift f¨ ur die gesamte Staatswissenschaft / Journal of Institutional and Theoretical Economics Bd. 121, H. 2. (April 1965), pp. 301-324. 93 [53] Stettner, L. (1982/83): Zero-Sum Markov Games With Stopping and Impul- sive Strategies. Appl. Math. Optim., 9, 1-24. [54] Strotz, R. (1955): Myopia and Inconsistency in Dynamic Utility Maximiza- tion. Review of Economic Studies 23 (1955), 165-180. [55] Touzi, N. and Vieille, N. (2002): Continuous-Time Dynkin games with Mixed Strategies. SIAM J. Control Optim., 41, 1073-1088. [56] Yong,J.(2011): ADeterministicLinearQuadraticTime-InconsistentOptimal ControlProblemMathematical Control and Related Fields, Volume 1, Number 1, March 2011., pp. 83-118 [57] Yong, J. : A Deterministic Time-Inconsistent Optimal Control Problem - an Essentially Cooperative Approach. Acta Appl. Math. Sinica, to appear. [58] Zhou, X. (2010): Mathematicalising Behavioural Finance. Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010. 94

Abstract (if available)

Abstract This dissertation consists of three parts. We first study the continuous time non-zero-sum Dynkin game which is a multi-player non-cooperative game on stopping times. We show that the Dynkin game has a Nash equilibrium point for general stochastic processes. The study extends the result of Hamadene and Zhang. ❧ The second part is to study the value dynamics of Dynkin game. In a zero-sum Dynkin game, where one player’s cost is the other player’s benefit, the value process is characterized by a two-barrier reflected backward stochastic differential equation, see Cvitanic and Karatzas. We build a parallel result for non-zero-sum Dynkin game and propose a new form of equilibrium called time-autonomous, which is mainly used to overcome non-uniqueness of the equilibrium. Under this framework, we construct the equivalency relation between a reflected BSDE system with jumps and non-zero-sum Dynkin Game. ❧ Finally we study Principal-Agent problem on stopping times, which addresses time-inconsistent issue in the sense that Bellman’s principle does not hold. We propose a method to solve Principal-Agent problem in discrete time framework.

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Asset Metadata

Creator Du, Jie (author)

Core Title On non-zero-sum stochastic game problems with stopping times

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 07/10/2012

Defense Date 05/14/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Dynkin game,non-zero-sum game,OAI-PMH Harvest,principal-agent problem,stopping time,time inconsistency

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Zhang, Jianfeng (committee chair), Ma, Jin (committee member), Zapatero, Fernando (committee member)

Creator Email dujiepku@gmail.com,jiedu@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-55813

Unique identifier UC11290344

Identifier usctheses-c3-55813 (legacy record id)

Legacy Identifier etd-DuJie-939.pdf

Dmrecord 55813

Document Type Dissertation

Rights Du, Jie

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA