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Dynamic approaches for some time inconsistent problems

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Dynamic approaches for some time inconsistent problems

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Content DYNAMIC APPROACHES FOR SOME TIME INCONSISTENT PROBLEMS by Chandrasekhar Karnam A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) May 2016 Copyright 2016 Chandrasekhar Karnam Dedicated To My Parents ii Acknowledgments First and foremost, I would like to express my deep gratitude towards both of my PhD advisers - Jin Ma and Jianfeng Zhang. It is a great privilege and an honor to work with both of them. I started my research at the end of year 2012 and since then, they have been constantly supporting me. They were always available whenever I need them. I am constantly amazed by their extensive knowledge in their elds (especially related to Stochastic Control, BSDE/FBSDEs). Moreover, they have a remarkable ability to teach their students. There have been many hard times during my PhD life and I am extremely indebted to my supervisors for their patience and moral support during those times. I sincerely thank for all the freedom they gave in the completion of the project. I am forever grateful to both of you - Prof. Jin Ma and Prof. Jianfeng Zhang. I owe many thanks to my undergraduate physics Professor Venkataraman Balakr- ishnan who inspired and motivated me to pursue PhD in Applied Mathematics. Although, I was majoring in Chemical Engineering for my undergraduate studies, I got attracted to beauty of Mathematics through his Theoretical Physics lectures. His teachings and lectures had a strong in uence on me. I am also grateful towards my Masters degree adviser - Prof. Francois Delarue, University of Nice. He introduced me to my PhD supervisors and he is one of the main reasons why I started my PhD at University of Southern California. My special thanks to Professors Yilmaz Kocer, Sergey Lototsky and Peter Baxendale for being part of my Oral/Dissertation committee. I also take this opportunity to thank my friends - Srivatsan, Nishan, Albert, Mohan, Ibrahim, Stephen, Christian and many others for their help and support during dierent periods of my studies. Last, but denitely not least, I would like to thank my parents. With out their love and support, I would not have come this far! Thank you very much. iii Table of Contents Dedication ii Acknowledgments iii Abstract vi Chapter 1: Utility Theory and DPP 1 1 Utility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Dynamic Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Choice Under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Stochastic Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Dynamic Programming Principle . . . . . . . . . . . . . . . . . . . . 7 Chapter 2: Time Inconsistency 9 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 Mean-Variance Optimization . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Optimal Exit Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Principal Agent Problem . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Deterministic Examples . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Approaches to handle Time Inconsistency . . . . . . . . . . . . . . . . . . . 16 Chapter 3: Problem Formulation 18 1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 BSDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1 Time Consistent Examples . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Time Inconsistent Examples . . . . . . . . . . . . . . . . . . . . . . . 24 4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.1 Expected Utility - BSDE formulation . . . . . . . . . . . . . . . . . 25 4.2 Mean Variance Optimization - BSDE formulation . . . . . . . . . . . 25 4.3 Optimal Exit Time - BSDE Formulation . . . . . . . . . . . . . . . . 26 4.4 Principal Agent Problem - BSDE formulation . . . . . . . . . . . . . 27 5 Novel Approaches for Time Inconsistency . . . . . . . . . . . . . . . . . . . 29 iv Chapter 4: Forward Utility Approach 30 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1 Mean-Variance Optimization . . . . . . . . . . . . . . . . . . . . . . 31 2.2 One Dimensional Example . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 Principal Agent Problem . . . . . . . . . . . . . . . . . . . . . . . . 33 3 Comparison Principle - Time Consistency . . . . . . . . . . . . . . . . . . . 33 3.1 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Existence of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Chapter 5: Master Equation Approach 43 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2 Dynamic Programming Principle . . . . . . . . . . . . . . . . . . . . . . . . 43 3 Time and Space Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1 Space Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Time Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.1 Illposed Master Equation . . . . . . . . . . . . . . . . . . . . . . . . 48 Chapter 6: Duality Approach 51 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2 Markovian Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.1 Geometric DPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.2 Auxiliary Control Problem . . . . . . . . . . . . . . . . . . . . . . . 52 2.3 Characterization of Reachable Sets . . . . . . . . . . . . . . . . . . . 53 3 Path Dependent/Non Markovian Case . . . . . . . . . . . . . . . . . . . . . 54 3.1 Geometric DPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2 Characterization of Reachable Sets . . . . . . . . . . . . . . . . . . . 56 Chapter 7: Existence of Optimal Control 58 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2 Introduction to Set Valued Analysis . . . . . . . . . . . . . . . . . . . . . . 58 2.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.2 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.3 Measurable Selection Theorem . . . . . . . . . . . . . . . . . . . . . 61 3 Backward Stochastic Dierential Inclusions . . . . . . . . . . . . . . . . . . 61 3.1 Existence and Compactness Results . . . . . . . . . . . . . . . . . . 62 4 Existence of Optimal Weak Solutions . . . . . . . . . . . . . . . . . . . . . . 63 Chapter 8: Conclusions 64 1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Chapter 9: Appendix 67 Bibliography 68 v Abstract In this thesis we discuss possible approaches to study time-inconsistent optimization prob- lems without the presumption that the optimal strategy exists. This immediately leads to the need to rene the concept of time-consistency, and invalidates any method that is based on Pontryagin's Maximum Principle. The main obstacle is the dilemma of having to study the Dynamic Programming Principle (DPP) in a situation where it is supposed to fail due to the nature of time-inconsistency. Our main innovation is to propose some type of \for- ward utility" under which the original time inconsistent problem (under the xed utility) becomes a time consistent one. We shall consider controlled multidimensional backward stochastic dierential equations as our model, which covers many existing time-inconsistent problems in the literature. One key observation is that the time inconsistency is essentially due to the failure of the comparison principle. We shall propose three approaches aiming at reviving the DPP for our model: the forward utility approach, the master equation approach, and the duality approach. Unlike the game approach in many existing works in continuous time models, all our approaches produce the same value as the original static problem. vi Chapter 1 Utility Theory and DPP We often observe that people change their preferences/choices over time. For example, an agent might prefer apples over oranges today and she might buy oranges today in the plan of eating them in couple of days. But later tomorrow, she might realize that she prefers oranges over apples. This kind of inconsistent preferences over time is referred as Time Inconsistency. Many problems arising in Economics, Finance and Social Sciences are Time - Inconsistent. In this chapter, we give mathematical foundations of preferences/choices for any agent. We usually represent the preferences using Utility functions. We also give a brief introduc- tion to Dynamic Programming Principle which is considered as one of the pillars of Control Theory. We also discuss the related issues of Time Consistency. In the next section, we brie y discuss the concepts of Choice, Preference, and Utility. We refer to [7] for details. 1 Utility Theory Microeconomic theory begins with modeling the choices made by any consumer/rm. At an abstract level, we have a setC of possible objects that might be chosen and an individual, the consumer who does the choosing. LetA be the collection of nonempty subsets of C from which the consumer can make a choice. We model the consumer choice by a choice function c :A7! 2 C such that c(A)A 8A2A. It is quite reasonable to assume that the choices made by the consumer in dierent situations are Coherent. For example, consider a student who opts for Mathematics when presented with Mathematics and Physics subjects. His choices would not be coherent if he chooses Physics when presented with Mathematics, Physics and Chemistry subjects. Now, let's formalize the denition of Choice Coherence Denition 1. A choice function c satises Choice Coherence if, for every pair x and y from X and A and B fromA, if x;y2A\B;x2c(A); and y = 2c(A); then y = 2c(B). Denition 2. A choice function c satises nite nonemptiness if c(A) is nonempty for every nite A2A. 1 The choices in general are formed by the consumer's preferences. A preference relation overC is binary relation which captures the choices made by consumer between any pair of objects in C . For every pair x and y from C , in general, we have exactly one of the following : xy (xy and yx) : The consumer prefers both x and y. xy (xy but not yx) : The consumer strictly prefers x over y. yx (yx but not xy) : The consumer strictly prefers y over x. Neither xy nor yx : The consumer does not prefer either x or y. However, if we drop the last possibility i.e. neither x y nor y x, we say is Complete. Moreover, it is very natural to assume that our preference relation satises transitivity. We formalize the denitions now. Denition 3. A preference relation, onC is complete if for everyx andy fromC either xy or yx or both. Denition 4. A preference relation, onC is transitive ifxy andyz impliesxz. We can characterize our choice function c as c(A) =fx2A :xy for all y2Ag for any A2A. Now, we would like to give a numerical representation to our preference relation using Utility function which is dened below. Denition 5. We call a function U :C7!R as Utility function if xy()U(x)U(y) At this point, one would naturally ask what assumptions on will guarantee the exis- tence of utility function. It is answered in the next theorem. Please refer to [7] for the proof. Theorem 1. (Existence of Utility function) Suppose is a complete and transitive relation onC . The relation can be represented by a utility function if and only if there exists some countable subsetC ofC such that if8x;y2C satisfying x y, we have x x y for some x 2C . 2 1.1 Dynamic Choice In this subsection, we would like to consider temporal aspects of choices and utility functions. The general problem consists of a consumer who must choose a consumption plan for the period [0;T ] (where 0<T <1) among various alternatives. LetC be the set of all integrable functions on [0;T ] whose lebesgue integral is equal to K. This can be thought as a budget constraint. If the preference relation onC satises the hypothesis of theorem (1), we can represent by utility functional 0 :C7!R. To make any interesting analysis, we must assume some structure on functional 0 . Refer [29] for the discussion of such assumptions. We assume that 0 is of the following form 0 (C) = Z T 0 (t)u(C(t);t)dt Here, u(C(t);t) is an \instantaneous utility function" and (t) is \time discounting func- tion". The consumer chooses the optimal plan C which satises 0 (C ) = sup C2C 0 (C) (1.1) Assuming u is dierentiable, the above problem can be solved easily using Calculus of variations as follows. Let y(t) = Z t 0 C(t)dt The problem then is to maximize : Z T 0 (t)u( _ y(t);t)dt subject to y(0) = 0;y(T ) = K. Using Euler-Lagrange equation, we get the necessary conditions for C as (t)(@ C u)(C (t);t) =I 0 for all 0tT (1.2) where I 0 is constant independent of time. This means that for optimal consumption plan, the discounted marginal utility is same for all times. 3 The consumption plan C is optimal starting at t = 0. However, the consumer after following the plan C till t =, may reevaluate his preferences and nd the best plan C ; for future tT according to (C ; ) = sup C2C (C) where (C) = R T (t)u(C(t);t)dt andC is collection of functions on [;T ] whose lebesgue integral is K R 0 C (t)dt. The necessary conditions for C ; may be found along the similar lines of (1.2) (t)(@ C u)(C ; (t);t) =I for all tT (1.3) In general, C ; (t)6= C (t) ( t T ) which means that the current optimal plan C ; is not consistent with past optimal plan C . This kind of change of preferences over time is referred as Time Inconsistency. However, if (t) = k t for some k > 0 we can see that C ; (t) = C (t) ( t T ) for any 2 (0;T ). This means that the consumer's optimal plan does not change even if he reevaluates at any future time. This kind of consistent preferences over time is referred as Time Consistency. One might be interested to know under what conditions, the problem (1.1) is time consistent. The following theorem states that only under \exponential discounting" the problem is time consistent. Refer to [29] for proof. Theorem 2. If (t) is dierentiable, then necessary and sucient conditions for problem 1.1 to be time consistent is that (t) =k t 0tT for some k> 0 1.2 Choice Under Uncertainty Most economic decisions often involve uncertain consequences. However, in some situations although the consumer cannot predict the outcome of his decision, he knows the probability distribution of possible outcomes. This is called objective uncertainty. On the other hand, if we don't know the probability distribution, we refer it as knightian uncertainty. 4 In this section, we assume that the choices available to the consumer are represented by probability distributions over a given space X; these distributions sometimes are called as lotteries or gambles. Hence, we are assuming objective uncertainty. LetP be the space of probability distributions over X. It's clear thatP is convex i.e. for any 0 a 1, P 1 ;P 2 2P, we have aP 1 + (1a)P 2 2P. The consumer's preferences are given by a preference relation on the spaceP. We seek a utility representation for the preference relation. As in the previous section, we need to make some rational assumptions about the preference relation to get any useful utility representation. Denition 6. We call a preference relation as von Neumann rational if it satises the following 4 properties. Completeness : For any 2 distributions P 1 ;P 2 2P, exactly one of the following holds P 1 P 2 ;P 2 P 1 ; or P 1 P 2 Transitivity : The preference is consistent across any 3 options. If P 1 P 2 and P 2 P 3 , then P 1 P 3 . Continuity :If P 1 P 2 P 3 , then there exists 0a 1 such that aP 1 + (1a)P 3 P 2 Independence : If P 1 P 2 , then for any P 3 2P and a2 (0; 1], aP 1 + (1a)P 3 aP 1 + (1a)P 2 Refer to chapter 5 in [7] for the proof of (3) and (4). Theorem 3. von Neumann-Morgenstern expected-utility : LetP be family of proba- bility distributions on X with nite support. A preference relation onP is von Neumann rational if and only if P 1 P 2 if and only if X x2supp(P 1 ) U(x)P 1 (x) X x2supp(P 2 ) U(x)P 2 (x) for some function U :X7!R which we call it as Utility function. 5 WhenP consists of distributions with innite support, the above theorem becomes technically challenging and we need further assumptions. WhenC =R d (orR d + ) , the most useful case for applications, we have the following theorem. Theorem 4. SupposeC =R d (or R d + ) andP is the space of Borel probability measures on C . If preference relation dened onP is von Neumann rational and also continuous in the weak topology, then there exists a bounded and continuous function U :C 7! R such that u(p) = Z C U(x)p(dx) represents. Remark 7. We can see that u(p) is E[U(X)] where X is a random variable which takes values inC and whose pushforward measure is p. Remark 8. If the consumer is risk averse, the utility function U is concave. 2 Stochastic Control Problem Now, we will make connections between Expected Utility and Stochastic Control problem. Assume that the consumer receives a reward X u T 2 R n at time T < 1. Here X u T is random and depends on the consumer's action/controlu. From theorem (4), if the consumer preferences are rational and continuous in weak topology, his behavior can be seen as if he is maximizing some expected utility. In this section, we will assume that X u T is a diusion process controlled by a process u and the utility function depends only on the terminal reward X u T with out any time-discounting. Let ( ;F;F;P) be a ltered probability space and B be a 1-dimensional Brownian motion adapted to F. The control u = (u s ) is a progressively measurable process with respect to F, valued in AR m . We assume that the state of system X (valued in R n ) is governed by the following SDE : dX s =b(X s ;u s )ds +(X s ;u s )dB s (2.4) The measurable functions b :R n A7!R n and :R n A7!R n satisfy the following uniform Lipschitz condition in A :9K 0;8x;y2R n ;8a2A; jb(x;a)b(y;a)j +j(x;a)(y;a)jKjxyj (2.5) 6 We denote byU the set of control processes u such that E Z T 0 jb(x 0 ;u t )j 2 +j(x 0 ;u t )j 2 dt <1 (2.6) where x 0 is any element in the support of X. By conditions (2.5) and (2.6), we know that for any u2U and for any initial condition (t;x)2 [0;T ]R n , there exists unique strong solution to (2.4) starting from x at time t. The result is quite standard and can be found in many texts. For example, refer to [18]. We denote this strong solution by n X t;x;u s ;tsT o and it is continuous a.s. We assume that utility function U :R n 7!R is measurable and nonnegative. For any givenu2U , we can then dene numerical representation of our preference on X u through the gain function J as follows : J(0;x 0 ;u) =E h U X 0;x 0 ;u T i (2.7) Maximizing expected utility reduces to the following 2 problems : Finding V 0 such that V 0 = sup u2U J(0;x 0 ;u) (2.8) and nding u such that J(0;x 0 ;u ) =V 0 (2.9) The problems (2.8) and (2.9) are dierent and in general, there may not exist any u . In this section, we focus on nding the value of V 0 with out reference to the existence of u . 2.1 Dynamic Programming Principle As discussed in the last section, the consumer may reevaluate his plan at any timet2 (0;T ). The problem then reduces to, v(t;x) := sup u2U J(t;x;u) (2.10) where J(t;x;u) =E h U X t;x;u T i (2.11) Our initial V 0 =v(0;x 0 ). Its also clear that v(T;x) =U(x). Dynamic Programming Principle (DPP) is a fundamental principle in the theory of stochastic control. DPP establishes relationships between v at dierent states. It sets up 7 an equation which gives the backward evolution ofv. For our problem, it can be formulated as follows : Theorem 5. (Dynamic Programming Principle) Let (t 1 ;x)2 [0;T ]R n and t 1 t 2 T . Then we have v(t 1 ;x) = sup u2U E h v t 2 ;X t 1 ;x;u t 2 i (2.12) Proof. First, lets prove that v(t 1 ;x) sup u2U E h v(t 2 ;X t 1 ;x; t 2 ) i : v(t 1 ;x) = sup u2U E h U X t 1 ;x;u T i = sup u2U E " U X t 2 ;X t 1 ;x;u [t 1 ;t 2 ] t 2 ;u [t 2 ;T] T !# As, v(t 2 ;X t 1 ;x;u t 2 ) = sup ~ u2U E U X t 2 ;X t 1 ;x;u t 2 ;~ u T , we have E " U X t 2 ;X t 1 ;x;u [t 1 ;t 2 ] t 2 ;u [t 2 ;T] T !# v t 2 ;X t 1 ;x;u [t 1 ;t 2 ] t 2 and hence, v(t 1 ;x) sup u2U E h v t 2 ;X t 1 ;x;u t 2 i (2.13) The proof of other inequality v(t 1 ;x) sup u2U E h v t 2 ;X t 1 ;x;u t 2 i (2.14) is bit more involved and it invokes measurable selection theorem. Refer to [2] for details. The innitesimal version of DPP will give rise to Hamilton Jacobi Bellman (HJB) partial dierential equation. Remark 9. Observe that in the proof of DPP, we did not assume the existence of optimal control u . Remark 10. It can be easily proved that if there exists optimal controlu then the problem is time consistent in the sense that u =u t; for any 0<t<T . 8 Chapter 2 Time Inconsistency In this chapter, we discuss Time Inconsistency issues related to stochastic/deterministic optimization problems. We provide several motivating examples of Time Inconsistency. 1 Introduction Let us consider a stochastic/deterministic optimization problem over a time interval [0;T ]: V 0 := sup u2U [0;T] J(u); (1.1) where U [0;T ] is an appropriate set of admissible controls dened on [0;T ], and J(u) is a certain utility (or cost functional) associated to the admissible control u. An admissible control u 2U [0;T ] is called \optimal" if J(u ) =V 0 , and for simplicity let us assume that the optimal control exists for the optimization problem (1.1). Now for xed 0<t<T , we consider the optimization problem over [t;T ]: V t := sup u2U [t;T] J t (u): (1.2) HereU [t;T ] denotes the set of admissible controls dened on [t;T ], andJ t is the utility over [t;T ]. If the original problem is stochastic,J t usually involves some conditional expectations, and it could be random. Thus, rigorously speaking, the supremum should be understood as essential supremum if the problem is stochastic optimization. Assume now that the optimization problem (1.2) also has an optimal control u t; . We say the problem is time-consistent if one can choose u t; in such a way that u t; s =u s ; tsT: (1.3) Clearly, the relation (1.3) amounts to saying that a (temporally) global optimum must be a local one, namely the \Bellman Principle" holds. An optimization problem (1.1) is called time-inconsistent if (1.3) fails to hold. Intuitively, time inconsistency means an optimal strategy planned today for tomorrow may not be optimal tomorrow. 9 2 Examples We now present several motivating examples of optimization problems that are time incon- sistent. 2.1 Mean-Variance Optimization One of the fundamental problems in portfolio selection is to nd optimal allocation of given wealth among a basket of securities. If we have to make a passive allocation of wealth (i.e. we just allocate our wealth at initial time and we do not alter our allocation until the end of the period), the Mean-Variance approach developed by Markowitz provides a fundamental basis. On the other hand, Expected Utility Theory (EUT) provides a framework for Dynamic active allocations or multiperiod portfolio selections. However, EUT has 2 problems from practical considerations : (i) It is often very dicult to know the utility function of investors (ii) The trade o between risk and return in utility theory is not explicit. In this subsection, we develop Mean-Variance approach for continuous time trad- ing/allocation. We will show that this problem exhibits Time Inconsistency feature. We will closely follow the model and solution developed in [23]. Let ( ;F;F;P) be a ltered probability space and W be a 1-dimensional Brownian motion adapted toF. For simplicity, we assume that the market has just one risky asset S (Stock) and one non risky asset P (Bond). We assume they have the following dynamics 8 < : dP t =rP t dt dS t =bS t dt +S t dW t (2.1) We assume that > 0. Consider an investor/agent who invests u t amount in the stock at any timet. Here,u t 2U [0;T ] :=L 2 F ([0;T ]). Assuming that he starts with x 0 amount, his wealth X will be governed by : 8 < : dX u t = [rX u t + t (br)]dt + t dW t X u 0 =x 0 (2.2) The objective of the investor is to maximize the expected terminal wealthE[X u T ] and at the same time minimize the variance of terminal wealth Var [X u T ] =E[(X u T ) 2 ] (E[X u T ]) 2 10 This leads to the following multi-objective optimization problem max u2U ((J 1 (u);J 2 (u))) (2.3) where J 1 (u) = E[X u T ], J 2 (u) =Var [X u T ] and U [0;T ] is an appropriate set of admissible controls dened on [0;T ]. Here, we seek to get Pareto Optimal controls. From the standard theory of multi- objective optimization problems, in our case a Pareto Optimal portfolio can be found by solving a single objective optimization problem where the objective is weighted average of J 1 andJ 2 . Hence, our original problem (2.3) can be solved by solving the following optimal control problem. max u2U (E[X u T ]Var [X u T ]) (2.4) The above problem due to the presence of the term (E[X u T ]) 2 in Var [X u T ] doesn't t in to theory we developed in rst chapter (1) for standard stochastic control problems. DPP fails and we need specialized tools for this problem. The above problem can be embedded in to a standard LQ problem. Using the solution to auxiliary LQ problem, we can nd a optimal control for the problem (2.4). Following section 4 from [23], we have the following optimal feedback control for the problem (2.4) u (t;x) = br 2 ( e rT x) (2.5) where =x 0 e rT + e ( br ) 2 T 2 For simplicity, assumeb = 1;r = 0 and = 1. Then the model (2.2) reduces to the following X u s =x 0 + Z s 0 u r dr + Z s 0 u r dB r ; s2 [0;T ]; u2U [0;T ] :=L 2 F ([0;T ]): (2.6) and the optimal control will be u (t;x) =x 0 x + e T 2 (2.7) By changing and reconsidering the problem (2.4) as V 0 := sup u2U [0;T] n cE[X u T ] 1 2 Var(X u T ) o : (2.8) 11 wherec> 0. We can easily see that the above optimization problem has an optimal feedback control: u (s;x) =x 0 x +ce T ; 0sT: (2.9) In other words, the optimal control is: u s = u (s;X ) = x 0 X s +ce T , s2 [0;T ], where X is the corresponding optimal dynamics satisfying X s =x 0 + Z s 0 [x 0 X r +ce T ]dr + Z s 0 [x 0 X r +ce T ]dB r ; s2 [0;T ]: Now let t2 (0;T ) be given, and we follow the control u on [0;t] so that X t is well- dened. Consider the optimization problem on [t;T ], starting from X t : X t;u s =X t + Z s t u r dr + Z s t u r dB r ; s2 [t;T ]; (2.10) and dene, similar to (2.8), the value of the optimization problem at time t: V t := esssup u2U [t;T] n cE t [X t;u T ] 1 2 Var t (X t;u T ) o ; (2.11) where E t [] = E[jF t ], and Var t is the conditional variance under E t . Again, as before we should have optimal control on [t;T ]: u t; (s;x) =X t x +ce Tt , s2 [t;T ]. It is not hard to see that, in general, u t; s =u t; (s;X t; s ) is dierent from u s =u (s;X s ) =x 0 X s +ce T for s2 [t;T ]. Thus the problem (2.6)-(2.8) is time inconsistent. Remark 11. Using Optimal control u for problem (2.6), we have E[X u T ] = c and Var[X u T ] = c 2 e T 1 . The loci of the points ( E[X u T ]x 0 x 0 ; Var[X u T ]) for dierent values of c is called Ecient Frontier. Remark 12. It is not very hard to see that u is also an optimal control for the following problem : V c 0 = inf fu:E[X u T ]=cg Var[X u T ]: (2.12) The above problem refers to minimizing Risk (Variance) for a given expected Reward. 2.2 Optimal Exit Time Expected Utility Theory (EUT) has been used by researchers for several decades as descrip- tive model of decision making under uncertainty. However, there has been substantial 12 evidence in the recent past that the EUT model does not provide an adequate description of how the choices are made by the decision maker. For example, the famous Allias Paradox can not be explained by EUT. Look at [19] for more details and examples. Kahneman and Tversky developed Cumulative Prospect Theory (CPT) as an alterna- tive to EUT which provides better description of individual choice. One of the principle ideas of CPT is Probability Distortion - most decision makers tend to overweight extreme (and unlikely) events but underweight the average events. In this subsection, we provide an example of time inconsistent optimal stopping strategy in the presence of probability distortion. We follow the model and results developed in [30] closely. Let ( ;F;F;P) be a ltered probability space and B be a 1-dimensional Brownian motion adapted toF. Consider a stock S which follows Geometric Brownian Motion 8 < : dS t =S t dB t S 0 =s (2.13) LetT be the set of allF stopping times withP( <1) = 1. The agent has to make a choice of2T and he will receive a payo U(S ). Also, the agent views the future on a distorted probability scale with distorting function w. We make the following assumptions on functions u and w. Assumption 13. U : [0;1)7! [0;1) is non-decreasing continuous function. w : [0; 1]7! [0; 1] is a strictly increasing, convex and absolutely continuous function with w(0) = 0 and w(1) = 1. Remark 14. The convexity ofw overweighs bad outcomes and underweighs good ones and hence captures the risk aversion of the agent. Look at [31] for details. The agent's target would be maximizing the distorted mean payo V 0 := sup 2T J() = sup 2T Z 1 0 w (P (U(S )>x))dx (2.14) Also, the agent would be interested in nding the optimal 2T such that J( ) =V 0 (2.15) The above problem can be solved and an explicit representation of is given in the following theorem. Refer to [30] for details and proof. 13 Theorem 6. Let function :R 2 7!R be dened as follows (a;b) = 1w sa ba U(a) +w sa ba U(b) If (a ;b ) = arg max 0<asb (a;b) (2.16) then (a ;b ) := 8 < : infft 0 :S t = 2 (a ;b )g; if a <b 0; if a =b (2.17) is an optimal stopping time to problem (2.15). The above theorem suggests that the optimal strategy for a risk averse agent is to hold the asset until it reaches one of the two thresholds, a andb . This strategy corresponds to the liquidation strategy widely used in stock trading namely \ Stop Loss or Take Prot". We can easily see that a and b depend on initial stock price s. Hence, our optimal strategy depends on the initial price. The agent might decide to reevaluate his strategy and solve the following problem at time t (0<t<T ) J( t; ) = sup f:t a.s.;2Tg J t () = sup f:t a.s.;2Tg Z 1 t w (P (U(S )>x))dx (2.18) Using theorem (6), we can write t; as an exit time from (a t; ;b t; ). It is not hard to see that, in general 6= t; . Thus the problem (2.14)-(2.15) is time inconsistent. 2.3 Principal Agent Problem In a special case of Holmstrom-Milgrom model, the Principal has to nd optimal contract C T assuming that Agent performs optimally given any contract. The model assumes expo- nential utilities for both Principal and Agent. It can be shown that this optimal contract to be devised by Principal changes as time evolves if we assume that Agents utility is independent of time. C T 6=C t; T (2.19) Look at section 4.4 of chapter 3 for problem formulation and details. 14 2.4 Deterministic Examples As all the examples provided so far involved randomness, the reader might think that Time Inconsistency is caused due to randomness in the problem. We note that the time inconsistency is due to the structure of the problem, not due to the randomness. We illustrate this point by a deterministic example in this subsection. Let T > 1,U [s;t] be the set of deterministic functions u : [s;t]! [0; 1], and Y 1;u t := Z T t [u s Y 2;u s ]ds; Y 2;u t := Z T t u s ds: (2.20) Consider the following optimization problem: for t2 [0;T ], V t := sup u2U [t;T] Y 1;u t : (2.21) By straightforward calculation, we have Y 1;u t = Z T t [u s Z T s u r dr]ds = Z T t [1 +ts]u s ds; (2.22) and then clearly the optimal control is: u t; s := 1 [t;(1+t)^T ] (s); tsT: (2.23) In particular, for 0<t<T 1, we see that u 0; s = 06= 1 =u t; s ; s2 (1; 1 +t): (2.24) That is, the problem (2.21) is time inconsistent. We now provide one more example : one dimensional deterministic and discrete time inconsistent problem. Our state at time t = 0; 1; 2 is given byy t . At any time, there are are 2 controls u 1 and u 2 . We consider the following backward dynamics. y 2 = 0; y u 1 1 = 1;y u 2 1 =2; y u 1 ;u 1 0 = 2;y u 2 ;u 1 0 =1;y u 1 ;u 2 0 =y u 2 ;u 2 0 = 0 (2.25) Consider the following optimization problem V 0 := max i=1;2;j=1;2 jy u i ;u j 0 j (2.26) 15 We can easily check that the optimal control for the above problem is (u 1 ;u 1 ). Now let us consider the dynamic version of the problem V 1 := max i=1;2 jy u i 1 j (2.27) Here, we see that (;u 2 ) is optimal control! Hence, the problem is Time Inconsistent. Remark 15. Note that the above problem doesn't admit Comparison Principle in the following sense :jy u 2 1 j>jy u 1 1 j butjy u 1 ;u 2 0 j<jy u 1 ;u 1 0 j. 3 Approaches to handle Time Inconsistency As it was pointed out in [29], there are typically two approaches for treating the time inconsistent problems: a) the strategy of precommitment, and b) the strategy of consistent planning. The former is to solve the static optimization problem (1.1), and then simply insist on using u (if it exists) throughout [0;T ], knowing that it may not be optimal anymore when t > 0. The latter one has developed into the popular \game approach" in the literature, in which the player plays with innitely many future selves. To illustrate the idea, let us consider the discrete time setting: 0 = t 0 <<t n =T . The \consistent planning" amounts to saying that at any t i , the player tries to nd optimal strategy u on [t i ;t i+1 ) by assuming the future selves have already found the optimal strategies and will actually use them on [t i+1 ;T ] = [t i+1 ;t i+2 )[[ [t n1 ;T ]. We note that an equilibrium in such a game approach should be similar to that of a principal agent problem, that is, in the sense of a sequential optimization problem, rather than a Nash equilibrium. The game approach makes sense in many applications but could be very challenging (as a game with uncountably many players!). Some successful applications of this approach in continuous time setting include, e.g., [3, 9, 17, 32], whereas [30] uses a precommitment strategy for a probability distortion problem. It is worth noting, however, that the game approach essentially is a \time consistent" method for an originally time-inconsistent prob- lem, but a slight drawback is that its value at t = 0 is, in general, dierent from the value V 0 in (1.1). In other words, the solution of the game problem, even if it exists, does not really solve the original static optimization problem (1.1) as far as the value V 0 is con- cerned. Moreover, if the equilibrium does not exist, which is quite possible in general, then it becomes unclear for one to even dene, much less study, its value. In our next chapters we suggest several possible approaches towards the same goal: to nd a \time-consistent" framework to study the value V 0 of the original static problem 16 (1.1), which is at least always well dened. We should note that the problem (1.1), or its \precommitment" nature, makes more sense in some applications. For example, in the so-called principal-agent problem (see section 4.4 of chapter 3), practically the principal cannot change the contract once it commenced, therefore one has to stick with the strategy designed at t = 0 for the whole contractual period. In fact, even without the practical consideration, solving problem (1.1) with a \time-consistent" method is interesting in its own right, mathematically. 17 Chapter 3 Problem Formulation As noted in the previous chapter, we call the following problem time consistent V 0 := sup u2U [0;T] J(u) =J(u ) (0.1) if u t; s =u s ; tsT: (0.2) for every 0<t<T , where u t; is the optimal control for the following problem : V t := sup u2U [t;T] J t (u) =J t (u t; ) (0.3) An optimization problem (0.1) is called time-inconsistent if (0.2) fails to hold for some 0<t<T . It is well-understood that there are typically two approaches for solving an optimization problem: DPP and the Stochastic Maximum Principle (SMP for short). The DPP mainly focuses on the dynamics of the value function, and it relies fundamentally on the time consistency; whereas SMP mainly focuses on the optimal control, and it does not require time consistency. Therefore, naturally, to date SMP has been used as the main tool for solving most of the time inconsistent problems, and the dynamical programming method has essentially been missing in the literature. A deeper question is thus, again, whether we say anything about the value V 0 without knowing the existence of the optimal control? To answer this question we quick to nd ourselves facing the dilemma: on the one hand the SMP is no longer relevant as it is only a necessary condition (therefore it is nullied if there is no optimal control); but on the other hand DPP does not make sense either, due to the lack of time-consistency. Our plan of attack is based on the following simple but interesting observation: many, if not all, time-inconsistent problems in the literature can be reformulated as the optimal control problems of multi-(even innite) dimensional backward SDEs (BSDE for short), and the time-inconsistency is essentially caused by the lack of the comparison principle for such BSDEs in general. (We shall present a few examples to illustrate this point in next section.) The properties of BSDEs then lead us to believe that 18 the time-inconsistent of the optimization problem could be (partially) due to the fact that we are using a xed criteria (or utility) throughout the time period [0;T ]. In fact, modulus some conditional expectation, the J t in (1.2) is essentially the same as the J in (0.1), which could be in con ict with the nature of the problem, and lead to time inconsistency. Therefore, inspired by the notion of forward utility (see, e.g., [25, 26]), we propose to nd some cost functional/utility J(t;u) for each t2 [0;T ] so that the optimization problem ~ V t := sup u2U [t;T] J(t;u) (0.4) becomes time consistent, and more importantly, we require that J(0;u) = J(u) so that ~ V 0 = V 0 . We remark that J(;) will be sought forwardly (in time), and ~ V will be solved backwardly. To better present our main idea, in this thesis we shall mainly focus on a general optimization problem for controlled multidimensional backward SDEs: 8 > > > < > > > : Y u t = + Z T t f(s;Y u s ;Z u s ;u s )ds Z T t Z u s dB s ; t2 [0;T ]; V 0 () := sup u2U [0;T] '(Y u 0 ): (0.5) We note that such a problem is typically time-inconsistent due to the lack of comparison theorem for the BSDEs involved, and, surprisingly, it covers many existing time-inconsistent problems in the literature (possibly in the form of forward-backward SDEs). Backward Stochastic Dierential Equations (BSDEs) have been studied extensively in the literature for past few decades. We record here some important results of BSDEs relevant for our purposes. For more details refer to [24]. Before doing so, we present our notation which we will use in the rest of this thesis. 1 Notation Throughout this thesis we consider a complete, ltered probability space ( ;F;P), on which is dened a 1-dimensional Brownian motion B = fB t g t2[0;T ] , where T > 0 denotes an arbitrarily xed time horizon. Let F =F B =fF B t g t0 be the natural ltration generated by B, with the standard augmentation. For a generic Euclidean space X, we denote its inner product by (x;y) = xy = x y, its norm byjxj = (x;x) 1=2 , and its Borel -eld by 19 B(X). If X = R dd , we denote A : B = trAB, for A;B2 X. Also, letGF be any sub--eld and [s;t] [0;T ], we denote L 2 (G;X) to be allX-valued,G-measurable random variable such thatkk 2 2 =E[jj 2 ]< 1. The inner product in L 2 (G;X) is denoted by (;) 2 =E[(;)], , 2L 2 (G;X). L 2 F ([s;t];X) to be allX-valued,F-adapted process on [s;t], such that kk 2;s;t =E Z t s j t j 2 dt 1=2 <1; In particular, ifX =R, we shall omitX in the above notations for simplicity. 2 BSDEs Let U R be a measurable set, and denote U := L 0 F ([0;T ];U). Consider the following controlled d-dimensional BSDE : for 2L 2 (F T ;R d ), Y u t = + Z T t f(s;Y u s ;Z u s ;u s )ds Z T t Z u s dB s ; t2 [0;T ]: (2.1) We make the following assumption on f : Assumption 16. The generatorf : [0;T ] R d R d U!R d is progressively measurable in all variables, uniformly Lipschitz continuous in (y;z), and E h Z T 0 sup u2U jf(t; 0; 0;u)jdt 2 i <1: Remark 17. Uncontrolled BSDE's (with out the termu inf) are covered in our framework by considering U to be singleton set. We now present the results if existence and uniqueness of (2.1). Theorem 7. (Existence and Uniqueness) : Given 2 L 2 (F T ;R d ) and f satisfying the assumption (16), for any given u 2 U there exists a unique solution (Y u ;Z u ) 2 L 2 F ([0;T ];R d )L 2 F ([0;T ];R d ) to the BSDE (2.1). We now state a very useful result known as Comparison Principle for BSDEs in 1 dimen- sion. Our understanding of Time Inconsistency of multi-dimensional controlled BSDE's crucially depends on this principle. For proof refer to [18]. 20 Theorem 8. Let 1 ; 2 2 L 2 (F T ;R) and the generator f satises the assumption (16) with d = 1. We x any u2U and let (Y 1 ;Z 1 ); (Y 2 ;Z 2 ) be the solutions to the BSDEs corresponding to ( 1 ;f) and ( 2 ;f) respectively. Suppose that 1 2 a.s. Then, Y 1 t Y 2 t for all 0tT a.s. : (2.2) Further, if P( 1 < 2 )> 0, then Y 1 0 <Y 2 0 . 3 Model As mentioned in the introduction of this chapter, we will focus on the following optimization problem : 8 > > > < > > > : Y u t = + Z T t f(s;Y u s ;Z u s ;u s )ds Z T t Z u s dB s ; t2 [0;T ]; V 0 () := sup u2U [0;T] '(Y u 0 ): (3.1) We note that hereY andZ ared-dimensional adapted processes and we make the following assumption on '. Assumption 18. The function ' :R d !R is continuous. Throughout this thesis the assumptions (16) and (18) will be our Standing Assumptions We shall refer to problem (3.1) as the Static Problem. We now consider the problem in a dynamic setting. For 0tT , we dene: V t () := esssup u2U '(Y u t ): (3.2) The above problem (3.2) is typically time inconsistent in the sense that the optimal control of static problem (3.1) is no longer optimal for the dynamic problem (3.2) over the time duration [t;T ]. However, we will rst provide 2 illustrative examples of the form (3.1) but which are Time Consistent. We will try to convince the reader that time consistency in both of these examples is due to Comparison Principle of BSDEs. 21 To facilitate our discussion let us introduce another notation. For any 0 t T , 2L 2 (F t ), and u2U , let (Y u (t;);Z u (t;)) denote the solution to the following BSDE on [0;t]: Y u s = + Z t s f(r;Y u r ;Z u r ;u r )dr Z t s Z u r dB r ; 0st: (3.3) Clearly, using the notationY u (;) and uniqueness of the solution to BSDE (3.3) we can write: Y u s =Y u s (t;Y u t ), 0stT ; and, in particular, Y u 0 =Y u 0 (t;Y u t ), t2 [0;T ]. 3.1 Time Consistent Examples Example 19. Assume that Assumption 16 is in force, and assume further that d = 1 and ' is increasing. Then, it is clear that the static problem (3.1) is equivalent to V 0 () :=' sup u2U Y u 0 . On the other hand, by the comparison principle of BSDEs and the monotonicity of', we see immediately that the dynamic problem (3.2) can also be written as: V t () ='(Y t ), 0tT , where f(s;!;y;z) := sup u2U f(s;!;y;z;u), and Y s = + Z T s f(s;Y s ;Z s )ds Z T s Z s dB s ; s2 [0;T ]: We claim that this problem is time consistent in the sense that the following DPP holds: V t 1 () = esssup u2U '(Y u t 1 (t 2 ;Y t 2 )); 0t 1 <t 2 T: (3.4) Indeed, for simplicity we set t 1 := 0 and t 2 := t. For any u2 U , we write Y u 0 = Y u 0 (t;Y u t ). By the comparison principle of BSDE and the monotonicity of ', we see that Y u t Y t =) Y u 0 =Y u 0 (t;Y u t )Y u 0 (t;Y t ) =) '(Y u 0 )'(Y u 0 (t;Y t )): (3.5) Since u is arbitrary, this implies that V 0 () sup u2U '(Y u 0 (t;Y t )): To see the opposite inequality, we assume rst that the there is a measurable function I : [0;T ] RR!U such that f(s;!;y;z;I(s;!;y;z)) =f(s;!;y;z); 8(s;!;y;z): (3.6) 22 This would particularly be the case whenU is compact andf is continuous inu, for instance. Then, clearly u s :=I(s;Y s ;Z s ) satises sup u2U '(Y u 0 (t;Y t )) ='(Y u 0 (t;Y t ))V 0 (); (3.7) namelyu is an optimal control for both (3.1) and (3.2). The proof of (3.7) in general case without (3.6) is a little more involved, which requires the measurable selection theorem. But since the argument is more or less standard, and it is irrelevant for our main purpose, we omit it. Consequently, (3.4) holds, and thus the problem is time-consistent. We should emphasize here that the DPP (3.4) does not require (3.6). However, if (3.6) does hold,thenu is optimal for both (3.1) and (3.2), which again indicates that the problem is time-consistent in terms of optimal control. The next example reinforces the importance of comparison principle for time consistency. Example 20. Letd 2. Consider the following multidimensional BSDE: for i = 1; ;d, Y i t = i + Z T t f i (s;Y s ;Z i s )ds Z T t Z i s dB s ; where f i (t;y;z i ) := sup u2U f i (t;y;z i ;u). Here we assumed that (i) for i = 1; ;d, f i does not depend on z j and is increasing in y j , for all j6=i; and (ii) ' is increasing in each components. Then it is well-known that the comparison principle remains true for such BSDEs. Thus we can follow the similar arguments as in Example 19 and show that V t () ='(Y t ), 0tT , and V t 1 () = esssup u2U '(Y u t 1 (t 2 ;Y t 2 )); P-a.s.; 0t 1 <t 2 T: Consequently, the problem is time consistent. Remark 21. From Example 19 and 20 we see that the comparison principle plays a crucial in establishing the DPP, whence the time consistency. In general, if ' is not monotone, or if comparison principle for BSDE fails, then the problem will become time inconsistent, as we will sew in the next section. 23 3.2 Time Inconsistent Examples One of the main reasons for the time inconsistency is the lack of comparison principle for the underlying stochastic dynamics. This is particularly the case when the stochastic dynamics involve a multidimensional BSDEs. In what follows we shall present an example showing that the time-inconsistency could happen even in one dimensional case as long as the cost functional involves the expectations nonlinearly (or non-monotonically). Example 22. Let U ad := L 2 F ([0;T ]; [1; 1]) and c2 R a constant. Consider a simple one-dimensional BSDE: Y u s =B T + Z T s u r dr Z T s Z u r dB r ; s2 [0;T ]; u2U ad ; (3.8) and, let '(y) :=jc +yj, y2R, we dene the optimal value by V 0 := sup u2U ad '(Y u 0 ) = sup u2U ad '(E[Y u 0 ]) = inf u2U ad c + Z T 0 E[u s ]ds : (3.9) u 2U ad is an optimal control if and only if: u s 1; if cT ; u s 1; if cT ; and Z T 0 E[u s ]ds =c; ifjcj<T: Now assume c =T . Let 0<t<T and consider the optimization problem on [t;T ]: V t := esssup u2U ad '(Y u t ) = essinf u2U ad T +B t + Z T t E t [u s ]ds ; (3.10) whereE t [] =E[jF t ]. Sincec =T , if the problem were time-consistent we would then expect thatu s =1, from the previous argument. However, we note that on the setfB t t2Tg, one has 0T +B t + (Tt)T +B t + Z T t E t [u s ]ds; for all u2U ad ; thus the optimal control for V t should be u t; s = 1 on the setfB t t 2Tg, instead of u s =1, a contradiction. Namely the problem (3.9) is time-inconsistent. 24 4 Applications Although studying the problem (3.1) is interesting mathematically by itself, however, we are motivated by the range of problems that can be reformulated as the optimal control problems of multi-(even innite) dimensional BSDEs or Forward Backward Stochastic Dierential Equations (FBSDEs). In this section, we provide several important problems which can be reformulated as (3.1). (Possibly involving FBSDE/dimension being innity). First, we start with well known time consistent problem of maximizing expected utility. 4.1 Expected Utility - BSDE formulation Example 23. As noted in rst chapter, the model for Expected Utility Theory has the following form : 8 > > < > > : X u t =x + Z t 0 b(X u s ;u s )ds + Z t 0 (X u s ;u s )dB s ; t2 [0;T ]; V 0 = sup u2U E [U(X u T )] (4.1) Please look at section 2 of chapter 1 for the assumptions on X;u; and g. The above problem can be reformulated easily in to controlled FBSDE as follows 8 > > > > > > < > > > > > > : X u t =x + Z t 0 b(X u s ;u s )ds + Z t 0 (X u s ;u s )dB s ; Y u t =U(X u T ) Z T t Z s dB s ; t2 [0;T ]; V 0 = sup u2U Y u 0 : (4.2) 4.2 Mean Variance Optimization - BSDE formulation Example 24. As noted in the section 2 of chapter 2, the mean variance optimization problem reduces to the following time inconsistent problem : 8 > > < > > : X u s =x 0 + Z s 0 u r dr + Z s 0 u r dB r ; s2 [0;T ]; u2U [0;T ] :=L 2 F ([0;T ]); V 0 := sup u2U [0;T] n cE[X u T ] 1 2 Var(X u T ) o : (4.3) 25 The above problem can be converted to an optimal control problem for multidimensional forward-backward SDEs. Indeed, we consider the following BSDEs: Y 1;u t =X u T Z T t Z 1;u s dB s ; Y 2;u t =jX u T j 2 Z T t Z 2;u s dB s ; t2 [0;T ]; (4.4) where X u satises the controlled stochastic dynamics (2.6). Then we can write (4.3) as V 0 := sup u2U '(Y 1;u 0 ;Y 2;u 0 ); (4.5) where'(y 1 ;y 2 ) :=cy 1 + 1 2 jy 1 j 2 1 2 y 2 . In other words, (2.6), (4.4), and (4.5) form an optimal control problem in which (4.4) is a 2-dimensional BSDE. Remark 25. Along the lines of the above example (24), we can rewrite the general problem of the form V 0 = sup u2U '(E [g(X u T )]) as V 0 = sup u2U '(Y u 0 ) where Y is the solution to a BSDE. 4.3 Optimal Exit Time - BSDE Formulation Example 26. Recall from section 2 of chapter 2, the problem of optimal exit time : 8 > > > < > > > : S t =s + Z t 0 S r dB r V 0 := sup 2T J() = sup 2T Z 1 0 w (P (U(S )>x))dx (4.6) The above problem can be written as optimal control problem Innite dimensional Forward BSDE (FBSDEs). For each x> 0 and 2T , consider the following FBSDE 8 > > > < > > > : S t =s + Z t 0 S r dB r Y ;x t = 1 U(S )>x Z T t Z ;x r dB r : (4.7) Then V 0 can be written as V 0 = sup Z 1 0 w (Y ;x 0 )dx (4.8) Hence, problem (4.7) - (4.8) forms an optimal control problem with innite dimensional FBSDEs for problem (4.6). 26 4.4 Principal Agent Problem - BSDE formulation Example 27. In this example we consider a special case of the Holmstrom-Milgrom model in the Pringcipal-agent Problem (cf. [6]). In this problem the principal is to nd the optimal contract assuming the agent(s) will always perform optimally given any contract. The main feature of principal's contract is that it is pre-committed, that is, it cannot be changed during a contractually designed duration. To be more precise, let A > 0; P > 0,R< 0 be constants, and consider two exponential utility functions: U A (x) := expf A xg; U P (x) := expf P xg: We denote the principal's control set by U P := L 2 (F T ), and the agent's control set by U A :=L 2 F ([0;T ]). Given any contract C T 2U P at t = 0, we consider the agent's problem: V A 0 (C T ) := sup u2U A E P u h U A C T 1 2 Z T 0 ju s j 2 ds i (4.9) where P u is a new probability measure dened by dP u dP := M u T := exp R T 0 u s ds 1 2 R T 0 ju s j 2 ds . We note that here the agent's control problem (4.9) is in a \weak for- mulation", and V A 0 (C T ) 0 is well-dened. We shall consider those contracts that satisfy the following \participation constraint" RV A 0 (C T ) 0; (4.10) where R< 0 is the \market value" of an agent that a principle has to consider at t = 0. It can be shown (cf. [6, Chapter 6]) that the agent's problem can be solved in terms of the following quadratic BSDE: Y A s =C T A 1 2 Z T s jZ A r j 2 dr Z T s Z A r dB r ; s2 [0;T ]: In fact, by a simple comparison argument for BSDEs one shows that the agent's optimal action is u =u (C T ) =Z A 2U A , with optimal value V A 0 =U A (Y A 0 ). Given the optimal u =u (C T ) we now consider the principal's problem: V P 0 := sup C T 2U P E P u [U P (B T C T )]; (4.11) 27 subject to the participation constraint (4.10). The solution to the problem (4.11)-(4.10) can be found explicitly (cf. [6, Chapter 6]). Indeed, the optimal contract is: C T := 1 A ln(R) +u B T + A 1 2 ju j 2 T; where u := 1+ P 1+ A + P is the corresponding agent's optimal action. We now consider the dynamic version of the agent's problem (4.9): for t2 [0;T ], V A t (C T ) := esssup u2U A E P u t h U A C T 1 2 Z T t ju s j 2 ds i ; (4.12) and the principle's problem, given agent's optimal control u(t;C T ): V P t := esssup C T 2U P E P u(t;C T ) t h U p (B T C T ) i ; (4.13) with the participation constraint V A t (C T )R. Solving the (constrained) principle's prob- lem (4.13) as before we see that the optimal contract is: C t; T := 1 A ln(R) +u (B T B t ) + A 1 2 ju j 2 (Tt); where u := 1+ P 1+ A + P . Clearly C t; T is dierent from C T , thus the problem is time- inconsistent. We note that the problem (4.11) can also be written as an optimal control problem for a (forward-)backward SDE. To see this, we rst note that by some straightforward arguments, one can show that for the optimal contract C T , the identity V 0 (C T ) =R must hold. Therefore we may impose a stronger participation constraint in (4.11): V 0 (C T ) =R, and rewrite Y A as a forward diusion: Y A s =U 1 A (R) + A 1 2 Z s 0 jZ A r j 2 dr + Z s 0 Z A r dB r ; s2 [0;T ]; which can be thought of as the optimal solution to the agent's problem (4.11) with dynamics Y A;u s :=U 1 A (R) + A 1 2 Z s 0 ju r j 2 dr + Z s 0 u r dB r ; s2 [0;T ]; (4.14) 28 with the relation C T = Y A T . Then, instead of viewing C T as the principal's control, we may view u :=Z A as the principal's control, and unify the principle-agent problem to the following optimization problem for multidimensional BSDEs: V 0 := sup u2U A Y P;u 0 ; (4.15) where (Y A;u ;Y P;u ) is the solution to the BSDEs (4.14) and Y P;u s =U P (B T Y A;u T ) + Z T s u r Z P;u r dr Z T s Z P;u r dB r ; s2 [0;T ]; (4.16) respectively. 5 Novel Approaches for Time Inconsistency By now, the reader might be very convinced that the problem (3.1) is very important and covers a large class of time inconsistent problems in the literature. Moreover, the time inconsistency of problem (3.1) is essentially due to the lack of Comparison Principle. In the next chapters, we propose 3 new approaches for handling time inconsistent problems : \Forward Utility" Approach \Master Equation" Approach \Duality" Approach. We discuss in detail all the above approaches in the next chapters. 29 Chapter 4 Forward Utility Approach 1 Introduction Let us recall our problem of interest : 8 > > > < > > > : Y u t = + Z T t f(s;Y u s ;Z u s ;u s )ds Z T t Z u s dB s ; t2 [0;T ]; V 0 () := sup u2U [0;T] '(Y u 0 ): (1.1) Also, note that the dynamic version of the above problem will be V t () := esssup u2U '(Y u t ): (1.2) As discussed in the previous chapters, the above problem is in general time inconsistent. The properties of BSDEs lead us to believe that the time-inconsistency in the above optimization problem could be (partially) due to the fact that we are using a xed criteria (or utility) ' throughout the time period [0;T ]. In fact, the utility for problem V t is same as the utility for problemV 0 , which could be in con ict with the nature of the problem, and lead to time inconsistency. Therefore, inspired by the notion of forward utility (see, e.g., [25, 26]), we propose to nd some (t;!;y) for each t2 [0;T ] so that the optimization problem ~ V t := sup u2U [t;T] (t;;Y u t ) (1.3) becomes time consistent, and more importantly, we require that (0;!;y) = '(y) so that ~ V 0 =V 0 . We remark that ~ V will be solved backwardly, and will be sought forwardly (in time). We refer to function as Forward Utility. Remark 28. Such a random function , if exists, has a spirit of the \forward utility" as those proposed in [25, 26], but it is fundamentally dierent in that the forward utility in [25, 26] essentially acts on eacht2 [0;T ] and optimizes over the time duration [0;t], whereas our random function acts on terminal timeT and optimizes over the time duration [t;T ]. 30 We now provide several examples in which we can nd and time consistent versions ( ~ V t ). 2 Examples 2.1 Mean-Variance Optimization Example 29. Let us recall the Mean-Variance Optimization problem which we discussed in chapters 2 and 3. 8 > > > > > > > > > > < > > > > > > > > > > : X u t =x 0 + Z t 0 u s ds + Z t 0 u s dB s ; Y 1;u t =X u T Z T t Z 1;u s dB s ; Y 2;u t =jX u T j 2 Z T t Z 2;u s dB s ; t2 [0;T ]; u2U [0;T ] :=L 2 F ([0;T ]); V 0 := sup u2U '(Y 1;u 0 ;Y 2;u 0 ): (2.1) where '(y 1 ;y 2 ) :=cy 1 + 1 2 jy 1 j 2 1 2 y 2 . The dynamic version of the above problem is V t := esssup u2U [t;T] '(Y 1;u t ;Y 2;u t ): (2.2) As discussed before, the above problem is time inconsistent. The optimal feedback control for the static problem V 0 is given by u (s;x) =x 0 x +ce T ; 0sT: (2.3) We now propose to nd time consistent version ~ V t for (2.2) by changing the utility ' with time. For any given t2 [0;T ], consider the following optimization problem ~ V t := sup u2U[t;T ] (t;;Y 1;u t ;Y 2;u t ): (2.4) where (t;!;y 1 ;y 2 ) := c t (!)y 1 + 1 2 jy 1 j 2 1 2 y 2 with c t = ce t e tT [X u t x 0 ] Using the similar arguments like in section 2 of chapter 2, we can nd the optimal control u t; for ~ V t : ~ u t; (s;x) =X t x +c t e Tt =x 0 x +ce T =u (s;x): 31 Hence the new problem (2.4) is time consistent. Remark 30. As noted in the remarks of section 2 of Chapter 2, we can view the problem (2.1) as minimizing the Risk for a given Expected Reward. The above method suggests that, we need to adjust our expected rewards (i.e. c t ) as time evolves, so that the problem will be time consistent. 2.2 One Dimensional Example Example 31. Let us reconsider the simple 1 dimesional BSDE example 22 of Chapter 3. 8 > > < > > : Y u s =B T + Z T s u r dr Z T s Z u r dB r ; s2 [0;T ]; u2U ad ; V 0 := sup u2U ad '(Y u 0 ): (2.5) where '(y) =jc +yj. The dynamic version of the above problem is V t := esssup u2U [t;T] '(Y u t ): (2.6) As discussed before, the above problem is time inconsistent. The control u 2U ad is an optimal control for static problem V 0 if and only if: u s 1; if cT ; u s 1; if cT ; and Z T 0 E[u s ]ds =c; ifjcj<T: For simplicity, let's chose c = T . Similar to the example in the previous subsection, if we allow the constant c in (2.5) to be time varying and even random, then the problem could become time consistent. Indeed, if we choosec t :=TtB t (and hencec 0 =c =T ), and consider ~ V t := esssup u2U ad (t;;Y u t ); where (t;!;y) :=jc t (!) +yj: (2.7) Then it is readily seen that ~ V t = essinf u2U ad (TtB t ) +B t + Z T t E t [u s ]ds = essinf u2U ad Tt + Z T t E t [u s ]ds ; and thus the optimal control is stillu =1. Hence, the ow n ~ V t o 0tT is time consistent. Moreover, ~ V 0 =V 0 . 32 2.3 Principal Agent Problem Example 32. Let us reconsider the last example (Principal Agent Problem - BSDE for- mulation) in section 4 of Chapter 3. As noted there, the optimal contract C T for static problem V 0 and the optimal contract C t; T for problem V t are dierent and the problem is time inconsistent. Again, the time-inconsistency can be removed if we allow the market value of the agents, the constant R, to be time varying (as it should be!). Indeed, if we set R t :=R exp A [u B t + A 1 2 ju j 2 t] ; and modify the participation constraint of the principal's problem as follows: V A t (C T )R t : (2.8) Then the optimal solution to the principle's problem with constraint (2.8) becomes: ~ C t; T = 1 A ln(R t ) +u (B T B t ) + A 1 2 ju j 2 (Tt) = 1 A ln(R) +u B T + A 1 2 ju j 2 T =C T : That is, the problem becomes time-consistent. 3 Comparison Principle - Time Consistency As pointed out in the previous section, when the original problem is time inconsistent, one can try to nd a certain random eld (t;!;y) satisfying (0;!;y) = '(y), such that a modied dynamic problem becomes time consistent: ~ V t () := esssup u2U (t;!;Y u t ): (3.1) However, in all the examples of previous section, the construction of the random eld (t;!;y) depends on the optimal control u of the static problem problem (for V 0 ), which is not desirable in general, especially in the cases when optimal control does not exist. The 33 main purpose of this chapter is to explore new ways to nd such random eld, so that (3.1) holds without having to have optimal control, and more importantly, we shall require that V 0 () = ~ V 0 (): (3.2) In other words, although we may not be able to nd the optimal control in the original (time- inconsistent) control problem, we nevertheless nd an alternative time-consistent problem that produce the same value. 3.1 Comparison Principle Our main goal here is to construct a random eld so that n ~ V t o 0tT is \time-consistent". And importantly, we want to construct in such a way that it is independent of optimal control of the static problem. As we advocated in previous chapters, the problem (1.2) is time inconsistent essentially due to lack of Comparison Principle. In this subsection, we dene a \weak comparison principle" which will be to used to construct independent of optimal control of the static problem. Hence, we give time consistent version of the problem (1.2). First, we dene the following Comparison Principle Denition 33. We say that a mapping : R d !R satises the comparison principle if for any t 1 <t 2 and any ; ~ 2L 2 (F t 2 ), esssup u (t 1 ;Y u t 1 (t 2 ;)) esssup u (t 1 ;Y u t 1 (t 2 ; ~ )); P-a.s. (3.3) whenever (t 2 ;) (t 2 ; ~ ), P-a.s. The following result justies our main idea. Please look at Chapter 6 for denitions of the setsN (t;!) andD(t;!) . Proposition 34. Assume that there exists a random eld satisfying the following prop- erties: (i) the mapping y7! (t;!;y) is continuous, for xed (t;!)2 [0;T ] ; (ii) (0;;y) ='(y), P-a.s.; and (iii) satises the comparison principle (3.3). 34 Then there exists Y 2L 2 (F) such that for t2 [0;T ], Y t 2N (t;), P-a.s., and 8 > < > : ~ V t := esssup u (t;;Y u t ) = (t;;Y t ); P-a.s. V 0 = ~ V 0 ='(Y 0 ): (3.4) Moreover, the following Dynamic Programming Principle holds: (t 1 ;;Y t 1 ) = esssup u (t 1 ;;Y u t 1 (t 2 ;Y t 2 )); P-a.s. (3.5) Proof. We rst recall from BSDE theory that for each (t;!)2 , the reachable setD(t;!) is bounded. Thus the setN (t;!), being closed, is compact. Then continuity of iny then leads to the existence of the maximum points for each (t;!)2 : Y t (!) := argmax y2N (t;!) (t;!;y)2N (t;!): (3.6) By measurable selection, we may assume Y is F-measurable. Then (3.4) follows directly from (3.11) and the continuity of . It remains to prove (3.5). Note that for any u 0 2U , (3.4) renders that (t 2 ;Y u 0 t 2 ) (t 2 ;Y t 2 ). Then, it follows from the comparison principle (3.3) that esssup u2U (t 1 ;Y u t 1 (t 2 ;Y u 0 t 2 ) esssup u2U (t 1 ;Y u t 1 (t 2 ;Y t 2 )); P-a.s. Note that by denitionY u t 1 (t 2 ;Y u 0 t 2 ) =Y u1 [0;t 2 ) +u 0 1 [t 2 ;T] t 1 , it follows immediately that (t 1 ;Y t 1 ) = esssup u2U (t 1 ;Y u t 1 ) esssup u2U (t 1 ;Y u t 1 (t 2 ;Y t 2 ))P-a.s. On the other hand, by (3.11), there existsfu n gU such that Y u n t 2 !Y t 2 , P-a.s. Now if we dene, for any u2U , ~ u n := u1 [0;t 2 ) +u n 1 [t 2 ;T ] , then by the continuity of and the standard stability result for BSDEs we have (t 1 ;Y u t 1 (t 2 ;Y t 2 )) = lim n!1 (t 1 ;Y u t 1 (t 2 ;Y u n t 2 )) = lim n!1 (t 1 ;Y ~ u n t 1 )) esssup u (t 1 ;Y u t 1 ) = (t 1 ;Y t 1 ): This completes the proof. 35 4 Existence of It is by now clear that in order to make the approach in this section work, we need to argue that the \forward utility" function satisfying the desired requirement does exist. This is a dicult in general, and should be a case by case task, depending on the specic applications. In the rest of the section we shall provide one illustrative example. Theorem 35. Assume that the coecients f and ' are of the following linear form: 8 > > > > > < > > > > > : f i (t;!;y;z;u) = d X j=1 [ i;j t (!)y j + i;j t (!)z j ] +c i (t;!;u); i = 1; ;d; '(y) = d X i=1 a i y i ; (4.7) where i;j 's and i;j 's are F-adapted processes, c i 's are F-progressively measurable random elds, and a i 's are constants. Then there exists a random eld satisfying the comparison principle (3.3), which takes the following linear form: (t;!;y) := d X i=1 A i t (!)y i ; with A i 0 =a i ; (4.8) where A i 's are F-adapted processes. Proof. We rst note that if d = 1, then the BSDE (2.1) is 1-dimensional, thus the compar- ison theorem holds. Further since ' is linear, whence monotone, thus the problem is time consistent and the theorem becomes trivial. We shall thus concentrate on multi-dimensional cases. Note also that for d 2, following an inductional arguments as illustrated in [21, Section 4.1], we need only prove the case d = 2. We shall split the proof (assuming d = 2) in three steps. Step 1. We begin by a heuristic argument which will lead us to the desired properties of the processesA 1 andA 2 . For convenience we shall assume thatA 1 andA 2 take the form of It^ o process: A i t =a i + Z t 0 b i s ds + Z t 0 i s dB s ; i = 1; 2; (4.9) 36 For any u2U and the corresponding solution (Y u ;Z u ), we dene Y u t := (t;;Y u t ) := 2 X i=1 A i t Y i;u t ; Z u t := 2 X i=1 [A i t Z i;u t + i t Y i;u t ]; t2 [0;T ]: (4.10) We hope to nd a pair of processes (A 1 ;A 2 ) so thatY u , along with an appropriately dened processZ u , satises a one dimensional BSDE, so as to reduce the problem to the cased = 1. To this end, we rst assume A 2 t a 2 6= 0, 0tT , Then, an easy application of It^ o's formula and some direct computations lead us to dY u t = h A 1 t dY 1;u t +Y 1;u t dA 1 t + 1 t Z 1;u t dt +a 2 dY 2;u t i (4.11) = h A 1 t 2 X j=1 [ 1;j t Y j;u t + 1;j t Z j;u t ] +A 1 t c 1 (t;u t ) +a 2 2 X j=1 [ 2;j t Y j;u t + 2;j t Z j;u t ] +a 2 c 2 (t;u t ) i dt [b 1 t Y 1;u t + 1 t Z 1;u t ]dt [A 1 t Z 1;u t + 1 t Y 1;u t +a 2 Z 2;u t ]dB t : Note that in this caseb 2 = 2 = 0, we see from (4.10) thatA 1 t Z 1;u t + 1 t Y 1;u t +a 2 Z 2;u t =Z u t , and thus Y 2;u t =a 1 2 [Y u t A 1 t Y 1;u t ]; Z 2;u t =a 1 2 [Z u t 1 t Y 1;u t A 1 t Z 1;u t ]: Plugging these into (4.11) and reorganizing terms yields: dY u t +Z u t dB t = h [A 1 t 1;1 t +a 2 2;1 t b 1 t ]Y 1;u t + [A 1 t 1;1 t +a 2 2;1 t 1 t ]Z 1;u t +A 1 t c 1 (t;u t ) +a 2 c 2 (t;u t ) +[A 1 t 1;2 t +a 2 2;2 t ]a 1 2 [Y u t A 1 t Y 1;u t ] + [A 1 t 1;2 t +a 2 2;2 t ]a 1 2 [Z u t 1 t Y 1;u t A 1 t Z 1;u t ] i dt = h a 1 2 [A 1 t 1;2 t +a 2 2;2 t ]Y u t +a 1 2 [A 1 t 1;2 t +a 2 2;2 t ]Z u t +A 1 t c 1 (t;u t ) +a 2 c 2 (t;u t ) + t Y 1;u t + t Z 1;u t i dt; (4.12) where t := A 1 t 1;1 t +a 2 2;1 t b 1 t ]a 1 2 A 1 t [A 1 t 1;2 t +a 2 2;2 t ]a 1 2 1 t [A 1 t 1;2 t +a 2 2;2 t ]; t := [A 1 t 1;1 t +a 2 2;1 t 1 t ]a 1 2 A 1 t [A 1 t 1;2 t +a 2 2;2 t ]: 37 Now setting t t 0, we see that (4.12) becomes a linear BSDE for (Y u ;Z u ). But this can be done by simply solving b 1 t := [A 1 t 1;1 t +a 2 2;1 t ]a 1 2 A 1 t [A 1 t 1;2 t +a 2 2;2 t ]a 1 2 1 t [A 1 t 1;2 t +a 2 2;2 t ]; 1 t := [A 1 t 1;1 t +a 2 2;1 t ]a 1 2 A 1 t [A 1 t 1;2 t +a 2 2;2 t ]: Note that the processes b 1 and 1 can be easily written as functions of the process a 1 2 A 1 by setting b 1 t =a 2 b 1 (t;!;a 1 2 A 1 t ) and 1 t =a 2 1 (t;!;a 1 2 A 1 t ), where b 1 (t;x) := j 1;2 t j 2 x 3 1;2 + 1;2 [ 1;1 2;2 ] 1;2 22 x 2 + 1;1 t 2;2 t 2;2 [ 1;1 t 2;2 t ] 1;2 t 2;1 t x + [ 2;1 2;1 t 2;2 t ]; (4.13) 1 (t;x) := 1;2 t jxj 2 + [ 1;1 t 2;2 t ]x + 2;1 t : Plugging this into (4.9), we obtain an SDE for A 1 t : A 1 t =a 2 =a 1 =a 2 + Z t 0 b 1 (s;a 1 2 A 1 s )ds + Z t 0 1 (s;a 1 2 A 1 s )dB s ; t 0: (4.14) We should note that since the coecients has quadratic growth in A 1 t and b has triple growth in A 1 t , the SDE (4.14) is a Ricatti equation in general sense and has only local solutions. However, if (4.14) is solvable, which we shall argue rigorously in the next step, then we have found the desired forward utility function (t;) =Y t . Step 2. We now substantiate the idea in Step 1 rigorously. If a 1 =a 2 = 0, then clearly V 0 () = 0 and there is nothing to prove. From now on we assume without loss of generality thatja 1 jja 2 j and a 2 6= 0. Denote 0 := 0. Recall (4.14) and consider the following SDE: A 1 t =a 1 =a 2 + Z t 0 b 1 s; [2]_A 1 s ^ 2 ds + Z t 0 1 s; [2]_A 1 s ^ 2 dB s ; t2 [0;T ]: (4.15) Clearly A 1 has global solution. Dene 1 := infft 0 :jA 1 t j 2g^T . Then A 1 t =a 1 =a 2 + Z t 0 b 1 s;A 1 s ds + Z t 0 1 s;A 1 s dB s ; 0 t 1 : (4.16) 38 We now set A 1 t := a 2 A 1 t and A 2 t := a 2 , for 0 t 1 . Then, noting thatjA 1 1 j = 2 (or j(A 1 1 ) 1 j = 1 2 ) when 1 <T and reversing the roles of A 1 and A 2 as in Step 1 we can then obtain coecients b 2 , 2 completely symmetric as those in (4.13), and an SDE on [ 1 ;T ]: A 2 t = (A 1 1 ) 1 + Z t 1 b 2 s; [2]_A 2 s ^ 2 ds + Z t 0 2 s; [2]_A 2 s ^ 2 dB s : Similarly A 2 has global solution, and that A 2 t = (A 1 1 ) 1 + Z t 1 b 2 s;A 2 s ds + Z t 0 2 s;A 2 s dB s ; 1 t 2 ; (4.17) where 2 := infft 1 :jA 2 t =A 1 1 j 2g^T . We then dene A 1 t := A 1 1 , and A 2 t := A 1 1 A 2 t , for 1 t 2 . Note that since A 1 1 A 2 1 = A 1 1 (A 1 1 ) 1 = a 2 = A 2 1 , both A 1 and A 2 are continuous at 1 . Now repeating the arguments, we may dene, for n 1, processesfA n g and stopping times 0 = 0 1 n , such that A 2n t = (A 2n1 2n1 ) 1 + Z t 2n1 b 2 s;A 2n s ds + Z t 2n1 2 s;A 2n s dB s ; 2n1 t 2n ; A 2n+1 t = (A 2n 2n ) 1 + Z t 2n b 1 s;A 2n+1 s ds + Z t 2n 1 s;A 2n+1 s dB s ; 2n t 2n+1 : Furthermore, for all n 1, it holds thatjA n t j < 2; n1 t < n , andjA n n j = 2 on f n < Tg. The rest of the argument will be based on the following fact, which will be validated in the next step: P [ n1 f n =Tg = 1: (4.18) Assuming (4.18), we can now dene continuous processes A 1 ;A 2 on [0;T ]: A 1 t :=A 1 2n1 ; A 2 t :=A 1 2n1 A 2n t ;; 2n1 t 2n ; A 1 t :=A 2 2n A 2n+1 t ; A 2 t :=A 2 2n ; 2n t 2n+1 : Now dene by (4.8) and (Y u ;Z u ) by (4.10). We can rewrite (4.11) as dY u t = h t Y u t + t Z u t + 2 X i=1 A i t c i (t;u t ) i dt +Z u t dB t ; 0tT; 39 where t = ( 1;2 t A 2n+1 t + 2;2 t ; on [ 2n ; 2n+1 ] 2;1 t A 2n t + 1;1 t ; on [ 2n1 ; 2n ]; (4.19) t = ( 1;2 t A 2n+1 t + 2;2 t ; on [ 2n ; 2n+1 ] 2;1 t A 2n t + 1;1 t ; on [ 2n1 ; 2n ]: Note thatjA 2n+1 t j 2 on 2n t 2n+1 andjA 2n t j 2 on 2n1 t 2n , both ; are bounded. Now denoting Y u t () to emphasize the dependence on the terminal condition , it follows from the denition (4.10) and the comparison of BSDEs that (T;) (T; ~ ) =) Y u t ()Y u t ( ~ ); 8u2U =) esssup u2U (t;Y u t (T;)) esssup u2U (t;Y u t (T; ~ )); P-a.s. The same argument can be used to treat any subinterval [t 1 ;t 2 ], proving (3.3). Step 3. It remains to prove (4.18). Fix some > 0. Note thatja 1 =a 2 j 1. By (4.15) and standard estimates for SDEs we can easily check that E sup 0tT jA 1 t j 2 C. Thus P( 1 <T^)P sup 0t jA 1 t j 2 P sup 0t jA 1 t A 1 0 j 1 E h sup 0t jA 1 t A 1 0 j 2 i CE h Z 0 jb 1 s; [2]_A 1 s ^ 2 j 2 ds + Z 0 j 1 s; [2]_A 1 s ^ 2 j 2 ds i C: Now setting := 1 2C , so that P( 1 <T; 1 ) 1 2 : (4.20) Similarly, noting thatjA 2 1 j = 1 2 andjA 2 2 j = 2 onf 2 <Tg, we have P 2 <T^ ( 1 +) F 1 1 2 : (4.21) Repeating the arguments, for any n one shows that P n+1 <T^ ( n +) F n 1 2 : (4.22) We shall prove (4.18) by arguing thatP S n1 f n =Tg c =P T n1 f n <Tg = 0. But since n 's are increasing, this amounts to saying that lim n!1 Pf n < Tg = 0. Now 40 for the given , we can assume that m < T (m + 1), for some m2N. We claim the following much stronger result, which is obviously implies (4.18): for any n 1, P( n <T ) (2n) m 2 n ; whenever m<T (m + 1): (4.23) We shall prove (4.23) by induction on m. First, if m = 0, namely 0<T, then P( n <T ) = P( n <T; 1 ) =P( 1 <T; 1 )P n <T F 1 ; 1 <T 1 2 P n <T F 1 ; 1 <T : thanks to (4.20). By (4.22), for k<n we have P n <T F k1 ; k1 <T 1 2 P n <T F k ; k <T : Then by induction we see that P( n <T ) 1 2 n1 P n <T F n1 ; n1 <T 1 2 n ; proving (4.23) for m = 0. Assume (4.23) holds for m 1 and we shall prove it for m. By (4.20) we have P( n <T ) = P( n <T; 1 ) +P( n <T; 1 >) P( 1 <T; 1 )P n <T F 1 ; 1 <T +P( n <T; n 1 <T) 1 2 P n <T F 1 ; 1 <T +P( n <T; n 1 <T): Note that (m 1)<Tm, then the inductional hypothesis implies that P( n <T; n 1 <T) (2n 2) m1 2 n1 ; and thus P( n <T ) 1 2 P n <T F 1 ; 1 <T + (2n 2) m1 2 n1 : By (4.22) , for k<n we have P n <T F k1 ; k1 <T 1 2 P n <T F k ; k <T + (2n 2k) m1 2 nk : 41 Then by induction we have P( n <T ) 1 2 n + n1 X k=1 (2k) m1 2 n1 = 1 + 2 P n1 k=1 (2k) m1 2 n : It is straightforward to check that 1 + 2 P n1 k=1 (2k) m1 (2n) m , proving (4.18), whence the theorem. 42 Chapter 5 Master Equation Approach 1 Introduction Having introduced time consistent versions for time-inconsistent control problems by modi- fying the utility functions, in this chapter we shall attack the value function itself. We begin by noticing that, unlike the forward stochastic control problem where the value function depends on the \initial data", in our problem the value V 0 () should be considered as the function of terminal data (T;). Our main idea is to let (T;) become \variables", and study the behavior of the value function. To be more precise, let us consider the following set A := n (t;) :t2 [0;T ];2L 2 (F t ) o [0;T ]L 2 (F T ): (1.1) We should note that the pair (t;)2A is \progressively measurable" in nature, that is, for each t, has to beF t -adapted. We now introduce a dynamic \value" function for our original problem. Let :A ! R be a real-valued function onA dened by (t;) = sup u2U '(Y u 0 (t;)); (t;)2A: (1.2) Clearly, it holds that (0;y) ='(y) and V 0 () = (T;): (1.3) 2 Dynamic Programming Principle We have several easy consequences for the value function . Among other things, we show that a \forward" dynamic programming principle actually holds without any extra conditions, even in such a time-inconsistent setting. Lemma 36. Assume that Assumption 16 is in force. Then, 43 (i) For each t, (t;) : L 2 (F t ) ! R is Lipschitz continuous, that is, there exists a constant C > 0, such that j (t; 1 ) (t; 2 )jCk 1 2 k L 2 (Ft) for any 1 ; 2 2L 2 (F t ): (2.1) (ii) satises the following \forward dynamic programming principle": for any 0 t 1 <t 2 T and any 2L 2 (F t 2 ), (t 2 ;) = sup u2U (t 1 ;Y u t 1 (t 2 ;)): (2.2) Proof. (i) For any 1 ; 2 2L 2 (F t ) and any u2U , by standard BSDE arguments we have jY u 0 (t; 1 )Y u 0 (t; 2 )j 2 CE[j 1 2 j 2 ]: This immediately leads to (2.1) since u2U is arbitrary. (ii) Let u2U be given. By the uniqueness of the BSDE we should have ' Y u 0 (t 2 ;) =' Y u 0 t 1 ;Y u t 1 (t 2 ;) t 1 ;Y u t 1 (t 2 ;) : Taking supremum over u we prove \" part of (2.2). To see the opposite inequality, we x an arbitrary u2U. For any "> 0, by the denition of , there exists u " 2U such that t 1 ;Y u t 1 (t 2 ;) ' Y u " 0 t 1 ;Y u t 1 (t 2 ;) +": Denote ~ u " :=u " 1 [0;t 1 ) +u1 [t 1 ;t 2 ) . Then by denition ofY we have Y u " 0 t 1 ;Y u t 1 (t 2 ;) =Y ~ u " 0 (t 2 ;) and thus t 1 ;Y u t 1 (t 2 ;) ' Y ~ u " 0 (t 2 ;) +" (t 2 ;) +": First taking supremum overu2U on both sides, and then letting" goes to zero we obtain the \" part of (2.2). This completes the proof. 44 3 Time and Space Derivatives With the essentially \free" dynamic programming equation (2.2), it is natural to envision an HJB-type equation for the value function . Before doing so, we need appropriate notions of time and space derivatives for . We propose some denitions of derivatives useful for our setting in the following subsections. 3.1 Space Derivatives First o, for each t2 [0;T ], viewing L 2 (F t ) as a Hilbert space, we can dene the spatial derivative as the standard Fr echet derivative: for any ; ~ 2L 2 (F t ), hD (t;); ~ i := lim "!0 (t; +"~ ) (t;) " ; (3.1) whenever the limit exists. We remark that, when D (t;) exists, it can (and will) be identied as a random variable inL 2 (F t ), thanks to the Riesz Representation Theorem. 3.2 Time Derivatives The temporal derivative, however, is much more involved. We rst note that the dynamic programming principle (2.2) is \forward", and more importantly, the value function is \progressive measurable", it is conceivable that there might be some dierences between two directional(forward and backward) time derivatives. As it turns out, if we use the right-derivative temporally (as one often does in the classical case), then the corresponding master equation will be illposed. We shall provide a detailed analysis on this point in Appendix. We will therefore use left-derivative. A simple-minded, albeit natural, denition of the left-temporal derivative can be as follows: D t (t;) := lim !0 (t;) (t;) : (3.2) However, bearing in mind the \progressive measurability" of (or the denition of the set A ), we see that2L 2 (F t ) is typically notF t -measurable, so (t;) may not even be well-dened. One natural choice to overcome this issue is to modify (3.2) to the following: lim !0 (t;) (t;E P t []) : (3.3) 45 However, although this dentition could actually be sucient for our purpose as we see in the study of backward path-dependent PDEs, it relies heavily on the underlying measure P, which would cause many unintended consequence when we encounter situations where various probability measures are involved, as we often see in applications. A universal, \measure-free", and potentially more applicable denition requires using \pathwise" Analysis. Look at Appendix for a brief primer on path derivatives. The left temporal derivative can be dened as follows : D t (t;) := lim !0 (t;) (t; t t ) ; (3.4) provided the limit exists. Here for 0 s t, t s (!) := (! s^ ), for each ! 2 := C([0;T ];R d ). We remark that, D t (t;) is a real number, if it exists. We have the following denitions of continuity, space and time derivatives. Denition 37. (i) 2C 0 (A ) if is continuous in (t;). (ii)2C 2 (F t ) (resp. C 2 b (F t )) if the induced process t 2C 2 ([0;t] ) (resp. C 2 b ([0;t] )). In this case, we denote @ t :=@ t t t ; @ ! :=@ ! t t ; @ 2 !! :=@ 2 !! t t : (3.5) (iii) 2 C 1 (A ) if 2 C 0 (A ), D exists and is in C 0 (A ), and for any t and any 2C 2 (F t ), D t (t;) exists. 4 Master Equation We note that there are two fundamental dierences between the current value function and the classical value function for forward stochastic problem : since the DPP is \forward", the Dierential Equation should also be a temporally forward; since the spatial variable in the value function is now a random variable in an L 2 space which is innite dimensional, the dierential equation is quite dierent from the traditional HJB equation (even those innite dimensional ones(!)). We therefore call the innitesimal version(dierential equation) of DPP as master equation, which seems to t the situation better than an \HJB equation". The main result of this chapter is the following theorem. 46 Theorem 38. Assume that Assumption 16 is in force. Assume further that 2 C 1 (A ) and f is bounded and continuous in all variables. Then, satises the following master equation onA : 8 < : D t (t;) = D (t;);@ t + 1 2 @ 2 !! + sup u2U D (t;);f(t;;@ ! ;u) ; (0;y) ='(y): (4.1) Proof. Fix 0<<t. We rst apply the functional It^ o formula (0.2) to get s = Z t s [@ t r + 1 2 @ 2 !! r ]dr Z t s @ ! r dB r ; tst: (4.2) Now for any u2U , let (Y u (t;);Z u (t;)) be the solution to BSDE (3.3). Denote Y u s :=Y u s (t;) s ; Z u s :=Z u s (t;)@ ! s ; tst: Then Y u s = Z t s h f(r;Y u r ;Z u r ;u r ) [@ t r + 1 2 @ 2 !! r ] i dr + Z t s Z u r dB r ; tst: (4.3) By standard BSDE estimates we see that E h sup tst jY u s j 2 + Z t t jZ u s j 2 ds i C 2 : We can now apply the forward dynamic programming principle (2.2) to get (t;) (t; t ) = sup u h t;Y u t (t;) (t; t ) i = sup u Z 1 0 D t; t +Y u t ; Y u t d: To identify the right hand side above, we rst deduce from (4.3) that I u := Y u t Z t t E t h f(s; s ;@ ! s ;u s ) [@ t s + 1 2 @ 2 !! s ] i ds = Z t t E t h f(s;Y u s ;Z u s ;u s )f(s; s ;@ ! s ;u s ) i ds: 47 HereE t [] :=E[jF t ]. Then, it is not hard to check, using Assumption 16, that E[jI u j 2 ] CE h Z t t [jY u s j 2 +jZ u s j 2 ]ds i C 3 : Consequently, as ! 0, we have (t;) (t; t ) = sup u Z 1 0 D t; t +Y u t d; Z t t E t h f(s; s ;@ ! s ;u s ) [@ t s + 1 2 @ 2 !! s ] i ds +I u = sup u D t; t ; Z t t E t h f(s; s ;@ ! s ;u s ) [@ t s + 1 2 @ 2 !! s ] i ds +o() = sup u D t; t ; Z t t h f(s; s ;@ ! s ;u s ) [@ t s + 1 2 @ 2 !! s ] i ds +o() = sup u D t; ; Z t t h f(t;;@ ! ;u s ) [@ t + 1 2 @ 2 !! ] i ds +o() = sup u D t; ; f(t;;@ ! ;u) [@ t + 1 2 @ 2 !! ] +o(): This implies (4.1) immediately. Remark 39. (i) From (4.1) we see that the master equation is a rst order (forward) path- dependent PDE (although it involves the second-order path-derivative of the state variable ). While this is obviously the consequence of the forward DPP (2.2) and our required initial condition on , it is also due to the fact that, for a forward problem, standing at t and looking \left", the problem is essentially \deterministic", hence the corresponding \HJB equation should be rst order. The left-temporal path derivative that we introduced in (3.4) is thus essential. (ii) The main diculty of this approach is the proper solution of the master equation (4.1). To the best of our knowledge, such an equation is completely new in the literature. Its wellposedness, in strong, weak, and viscosity sense, seem to be all open at this point. We hope to be able to address some of them in 4.1 Illposed Master Equation We have emphasized in the previous section the importance for using the left-temporal derivative, given the fact that satises a forward dynamic programming principle. In what follows we shall reinforce this point by explaining how a \traditional" right-temporal derivative could actually lead to a ill-posed master equation. 48 Let us suppose that we dene a time-derivative in the following traditional way: D + t (t;) := lim #0 (t +;) (t;) ; (t;)2 ; (4.4) whenever the limit exists. Since by our denition of A , for each > 0, 2 L 2 (F t ) L 2 (F t+ ), thus (t +;), whence D + t (t;), is well-dened for all (t;)2A . Now let us check the equation that such a derivative will produce. Again, by DPP (2.2) we have (t +;) (t;) = sup u h (t;Y u t (t +;)) (t;) i = sup u Z 1 0 D (t; +Y u t );Y u t d (4.5) where Y u s :=Y u s (t +;), t s t +. Note that, if we denote Z u s :=Z u s (t +;), then (Y u ;Z u ) satises the BSDE: Y u s = Z t+ s f(r; +Y u r ;Z u r ;u r )dr Z t+ s Z u r dB r ; tst +: Then, the standard BSDE estimates would tell us that, E h sup tst+ jY u s j 2 + Z t+ t jZ u s j 2 ds i C 2 : Again, let us denote I u := Y u t E t h Z t+ t f(s;; 0;u s )ds i : Then, assuming Assumption 16 we have jI u j = E t h Z t+ t [f(s; +Y u s ;Z u s ;u s )f(s;; 0;u s )]ds CE t h Z t+ t [jY u s j +jZ u s j]ds i ; and consequently E[jI u j 2 ]CE h Z t+ t [jY u s j 2 +jZ u s j 2 i dsC 3 : 49 Now (4.5) will lead to that (t +;) (t;) = sup u D (t;);E t h Z t+ t f(s;; 0;u s )ds i +o() = sup u2L 2 (Ft) D (t;);f(t;; 0;u) +o(): In other words, we will arrive at the following rst order PDE: 8 > < > : D + t (t;) = sup u2L 2 (Ft) D (t;);f(t;; 0;u) ; t> 0; (0;y) ='(y): (4.6) We remark that the equation (4.6) is typically ill-posed. Indeed, (4.6) involves only f(;; 0;), while the dened in (1.2) obviously depends on f(;;z;). So unless the function f is independent of the variable z, there is essentially no hope that the equation (4.6) will have a unique solution, as the value functions of two completely dierent opti- mization problems can satisfy the same master equation(!). We therefore conclude that D t , not D + t , is the right choice of temporal derivative for the master equation. 50 Chapter 6 Duality Approach 1 Introduction Let us recall our problem of interest : 8 > > > < > > > : Y u t = + Z T t f(s;Y u s ;Z u s ;u s )ds Z T t Z u s dB s ; t2 [0;T ]; V 0 () := sup u2U [0;T] '(Y u 0 ): (1.1) In this chapter we present an approach that is simple but quite eective if one focuses only on nding the value of the above static problem (1.1) with out reference to V t . To illustrate the idea better we begin by considering the Markovian case. 2 Markovian Case We assume that in BSDE (1.1), =g(B T ) and f =f(t;B t ;y;z;u). We shall give heuristic arguments in this section, and give complete proofs for the general non-Markovian path- dependent case in the next section. To begin with, for each (t;x)2 [0;T ]R d , we denote B t;x s :=x +B s B t , st, and B t s :=B t;0 s . Consider the set D(t;x) := n y2R d :9Z2L 2 F ([0;T ]);u2U; s.t. X t;x;y;Z;u T =g(B t;x T ); P-a.s. o ; (2.1) where X t;x;y;Z;u denotes the solution to the forward SDE: X s = y Z s t f(r;B t;x r ;X r ;Z r ;u r )dr + Z s t Z r dB t;x r (2.2) = y Z s t f(r;B t;x r ;X r ;Z r ;u r )dr + Z s t Z r dB r ; tsT: (2.3) 51 Clearly,X can be thought of as a forward version of the solution to the BSDE (2.1) on [t;T ], and the setD(t;x) is simply the reacheable setfY u t ;u2Ug given B t = x. In particular, D(0; 0) =fY u 0 :u2Ug, and our original optimization can be written as V 0 () = sup y2D(0;0) '(y): (2.4) It is worth noting that sup y2D(0;0) '(y) in (2.4) is a nite dimensional constrained opti- mization problem. So the value V 0 () could be determined rather easily, provided one can characterize the setD(0; 0), which we now describe. 2.1 Geometric DPP Observing thatD(T;x) =fg(x)g, we shall viewfD(t;)g 0tT as a backward, set-valued dynamic system, and we claim that the familyD(;) satises the following geometric DPP: for any 0t 1 <t 2 T , D(t 1 ;x) := n y2R d :9(Z;u) such that X t 1 ;x;y;Z;u t 2 2D(t 2 ;B t 1 ;x t 2 ); P-a.s. o : (2.5) We note that given the geometric DPP we can now nd a time-consistent optimization problem that will lead to the original value V 0 (). To this end, we borrow the idea of the method of optimal control for solving a forward-backward SDE (cf. [24]). 2.2 Auxiliary Control Problem Consider the following auxiliary control problem: W (t;x;y) := inf Z;u E n X t;x;y;Z;u T g(B t;x T ) 2 o : (2.6) Clearly, (2.6) is a standard stochastic control problem, and it is well-known that W should be the (unique) viscosity solution to the following (degenerate) HJB equation: 8 > < > : @ t W + 1 2 @ 2 xx W + inf z;u n [ 1 2 @ 2 yy W : (zz ) +@ 2 xy W :z@ y Wf(t;x;y;z;u) o = 0; W (T;x;y) =jyg(x)j 2 : (2.7) Remark 40. The HJB equation (2.7) can be simplied slightly when f is independent of z. In fact, assuming all variables are 1-dimensional, one can separate the variables u andz in (2.7), and minimize the quadratic form (for z) to obtain, at least formally, that 52 8 > > > > > > > < > > > > > > > : 0 =@ t W + 1 2 @ 2 xx W 1 2 [@ 2 xy W ] 2 @ 2 yy W + inf u n @ y Wf(t;x;y;u) o =@ t W + 1 2 det[D 2 W ] @ 2 yy W + inf u n @ y Wf(t;x;y;u) o ; t2 [0;T ); W (T;x;y) =jyg(x)j 2 ; (2.8) where D 2 W denotes the Hessian of W , and \det" means the determinant. This is an HJB equation of parabolic Monge-Amp ere type, whose numerical solutions have been studied extensively (cf. e.g., [28]). 2.3 Characterization of Reachable Sets By denition (2.1), it is clear that W (t;x;y) = 0 whenever y2D(t;x). More generally, we expect and will show that, for any (t;x), the following relationship between the setD(t;x) and the \nodal set" of the function W holds: N (t;x) := n y2R d :W (t;x;y) = 0 o = D(t;x): (2.9) whereD(t;x) denotes the closure ofD(t;x). Then (2.4) amounts to saying that V 0 () = sup y2D(0;0) '(y) = sup y2N (0;0) '(y): (2.10) In other words, we have characterized the set D(0; 0) in terms ofN (0; 0), the nodal set of W , which in theory is a much benign task to deal with (for example, numerically). We shall refer to this method as the \duality approach". Remark 41. We remark that the idea of \duality" is not new (see, for example, [1, 14, 15, 27]). The main dierence, however, is the temporal direction of the \reachable set"D(;). More precisely, unlike the standard literature where the controlled state process is forward (hence so is the corresponding reachable setsD(;)), in our case the reachable sets moves backwardly, and consequently the geometric DPP (2.5), as well as the PDE characterization (2.9) become possible. 53 3 Path Dependent/Non Markovian Case The arguments in the above section can be extended to a more general non-Markovian setting for which the corresponding HJB equation (2.7) will be in the form of the so-called path dependent PDEs (cf. e.g., [10, 11, 12]). Let us recall some basic notions of functional It^ o calculus (Please look at appendix and references there in for details). We assume that all the discussions below are in the canonical space =C 0 ([0;T ];R d ). For xed (t;!)2 , we denoteP t;! [ ] =P[jF t ](!) be the regular conditional probability givenF t . It is then clear that underP t;! , any ^ !2 can be decomposed into the concatenation: ^ ! :=! t ~ ! =!1 [0;t] + (~ ! +!(t))1 [t;T ] , ~ !2 t := C 0 ([t;T ];R d ). Consequently, for 2 L 0 ( ), we denote t;! (~ !) := (! t ~ !), for P t;! -a.s. ~ !2 t . We shall make use of the following extra assumption: Assumption 42. The mapping (t;!) 7! f(t;!;y;z;u) belongs to C 0 (), for all xed (y;z;u); and the mapping !7!(!) is continuous underkk T , the norm of . 3.1 Geometric DPP Using the path-notations dened above we now dene, similar to the Markovian case, that for xed (t;!)2 , D(t;!) := n y2R d :9(Z;u); s.t. X t;!;y;Z;u T = t;! ; P t 0 -a.s. o ; (3.1) where X t;!;y;Z;u is the solution to the following forward SDE, under P t;! : X s =y Z s t f t;! (r;;X r ;Z r ;u r )dr + Z s t Z r dB t r ; tsT: (3.2) Here the function f t;! (r; ~ !;y;z;u), (r; ~ !)2 [t;T ] t is dened the same as t;! explained before. We should note that, by the denition of d 1 and Assumption 42, this means that we are xing the path of ^ ! as ! up to time t in f. In this section, we discuss geometric DPP for the set valued processD(t;!) dened by (3.1), in the spirit of Soner & Touzi [27]. Intuitively, we expect: D(t 1 ;!) = n y2R d 0 :9(Z;u)2L 2 (F t 1 ;R d 0 d )U t 1 such that (3.3) X t 1 ;!;y;Z;u t 2 2D(t 2 ;! t B t 1 ); P t 1 0 -a.s. o ; 0t 1 <t 2 T: 54 Denote byD 0 (t 1 ;!) the right side of (3.3). One can easily proveD(t 1 ;!) D 0 (t 1 ;!). However, we have some technical diculty to prove the opposite inclusion. We thus prove a weaker version of geometric DPP. Recall (3.11) and dene, for any "> 0, N " (t;!) :=fy2R d 0 :W (t;!;y)"g: (3.4) It is clear that N (t;!) =\ ">0 N " (t;!): (3.5) Theorem 43. Under Assumptions 16 and 42, the following geometric DPP holds true: N (t 1 ;!) := \ ">0 n y2R d 0 :9(Z " ;u " )2L 2 (F t 1 ;R d 0 d )U t 1 such that (3.6) X t 1 ;!;y;Z " ;u " t 2 (~ !)2N " (t 2 ;! t ~ !); P t 1 0 -a.e. ~ !2 t 1 o ; 0t 1 <t 2 T: Proof. LetN 0 (t 1 ;!) denote the right side of (3.6). For simplicity, we assume t 1 = 0 and t 2 =t, and note that ! 0 = 0, so we shall only prove N (0; 0) =N 0 (0; 0) := \ ">0 n y2R d 0 :9(Z " ;u " )2L 2 (F;R d 0 d )U such that X 0;0;y;Z " ;u " t (!)2N " (t;!); P 0 -a.e. !2 o : (3.7) Under the assumptions and following the arguments in [10], W is uniformly continuous in (t;!;y), with certain modulus of continuity function , and satises the following DPP: W (0; 0;y) = inf (Z;u)2L 2 (F;R d 0 d )U E P 0 h W (t;B ;X 0;0;y;Z;u t ) i : (3.8) First, let y2N 0 (0; 0). For any "> 0, let (Z " ;u " ) be as in the right side of (3.7). Then W (t;B;X 0;0;y;Z " ;u " t ) " P 0 -a.s. and thus E P 0 h W (t;B;X 0;0;y;Z " ;u " t ) i ". This, together with (3.8), implies that W (0; 0;y) = 0. Then y2N (0; 0) and henceN 0 (0; 0)N (0; 0). To see the opposite inclusion, let y2N (0; 0). Then for any " > 0, there exists y " 2 D(0; 0) such thatjy " yj". By (3.1) there exists (Z " ;u " )2L 2 (F;R d 0 d )U such that X 0;0;y";Z " ;u " T =,P 0 -a.s. It is straightforward to see that, forP 0 -a.e. !2 , Z ";t;! 2L 2 (F t ;R d 0 d ); u ";t;! 2U t ; (X 0;0;y";Z " ;u " s ) t;! =X t;!;X 0;0;y";Z " ;u " t ;Z ";t;! ;u ";t;! s ;tsT; P t 0 -a.s. 55 Then X t;!;X 0;0;y";Z " ;u " t ;Z ";t;! ;u ";t;! T = (X 0;0;y";Z " ;u " T ) t;! = t;! and thus X 0;0;y";Z " ;u " t 2D(t;!). Denote X :=X 0;0;y";Z " ;u " X 0;0;y;Z " ;u " . Then X s =y " y + Z t 0 r X r dr; 0st; where is bounded, thanks to the Lipschitz continuity of f in y. Then clearlyjX t j Cjy " yjC", and thus jW (t;!;X 0;0;y;Z " ;u " t (!))j = jW (t;!;X 0;0;y;Z " ;u " t (!))W (t;!;X 0;0;y";Z " ;u " t (!))j (jX t (!)j)(C"): This implies that X 0;0;y;Z " ;u " t (!)2N (C") (t;!). Since " > 0 is arbitrary, we obtain y2 N 0 (0; 0), and thusN (0; 0)N 0 (0; 0). 3.2 Characterization of Reachable Sets A more important element in this scheme, however, is the characterization of the setD(t;!) in terms of the nodal set of the solution of a degenerate parabolic PDE. To this end, we introduce an auxiliary control problem: W (t;!;y) := inf Z;u E P t;! 0 h X t;!;y;Z;u T t;! 2 i : (3.9) This is a non-Markovian control problem in a path-dependent setting. Applying the result of [10, 11] we conclude that W is the unique viscosity solution to the following degenerate backward path-dependent HJB equation: 8 < : @ t W + 1 2 @ 2 !! W + inf z;u h 1 2 @ 2 yy W : (zz ) +@ 2 !y Wz@ y Wf(t;!;y;z;u) i = 0; W (T;!;y) =jy(!)j 2 : (3.10) The following result substantiates the main idea of the duality approach. Theorem 44. Under Assumptions 16 and 42, for any (t;!) we have N (t;!) := n y2R d :W (t;!;y) = 0 o = D(t;!): (3.11) 56 Consequently, V 0 () = sup y2N (0;0) '(y): (3.12) Proof. We begin by noting that (3.12) is a direct consequence of (3.11) as well as the continuity of '. So it suces to prove (3.11). First, we x (t;!)2 and lety2D(t;!). By denition there exists (Z;u)2L 2 F ([t;T ]) U ad ([t;T ]) such that X t;!;y;Z;u T = t;! ,P t;! 0 -a.s. Then we must have W (t;!;y)E P t;! 0 h X t;!;y;Z;u T t;! 2 i = 0: That is, y2N (t;!) and consequentlyD(t;!)N (t;!). Moreover, the continuity of W in y then renders that N(t;!) is a closed set, which leads to thatD(t;!)N (t;!). Conversely, if y2 N (t;!), then by denition for any " > 0, there exists (Z " ;u " )2 L 2 F ([t;T ])U ad ([t;T ]) such that E P t;! 0 h X t;!;y;Z " ;u " T t;! 2 i ": (3.13) Denote t;! " := X t;!;y;Z " ;u " T . SinceY u " t (T; t;! " ) = y, P t;! 0 -a.s., by the standard BSDE esti- mates we have, for the given (t;!)2 , jY u " t (!)yj 2 = Y u " 0 (T; t;! )Y u " 0 (T; t;! " ) 2 CE P t;! 0 h t;! " t;! 2 i C": Since Y u " t (!)2D(t;!) and " is arbitrary, we see that y2D(t;!). We remark again that the argument above relies fundamentally on the estimates for BSDEs, for which the path-dependence is irrelevant. 57 Chapter 7 Existence of Optimal Control 1 Introduction Let us recall our problem of interest : 8 > > > < > > > : Y u t = + Z T t f(s;Y u s ;Z u s ;u s )ds Z T t Z u s dB s ; t2 [0;T ]; V 0 () := sup u2U [0;T] '(Y u 0 ): (1.1) There are 3 important questions related to this problem : Question 1 : To determine optimal cost/prot V 0 . When d = 1 this can be done by using DPP. But, when d > 1 and ' non-monotone, the problem is time inconsistent and we don't have any immediate DPP. However, using the methods developed in chapter 6 of this thesis, we can give Geometric DPP and write a PDE to get the static value V 0 . Question 2 : To determine the dynamics ~ V t so that dynamic problem is time- consistent. This question can be addressed using the methods developed in chapters 4 and 5. Question 3 : In which cases there exist optimal controls? We are going address this question in this chapter. We will use set valued approach and measurable selection theorems will be key in this approach. We now provide a brief introduction to Set Valued Analysis. For details look at [1] 2 Introduction to Set Valued Analysis Let (X;kk X ) be a separable Banach Space,K b c (X ) the family of all nonempty, closed, bounded and convex subsets ofX . The Hausdor distance H X onK b c (X ) is dened by H X (A;B) = maxfsup a2A dist X (a;B); sup b2B dist X (b;A)g (2.1) 58 where dist X (a;B) = inf b2B kabk X . It is known that (K b c (X );H X ) is a complete and separable metric space. In many applications,K b c (X ) valued maps may be too restrictive and we can consider a bigger class of sets Cl(X ) which denote the class of all nonempty closed subsets ofX . By set-valued mapping (or multifunction) we mean any function F which takes values in Cl(X ) (orK b c (X )). Let (M;M;) be any measure space. We have following denitions of Measurability and Integrably Bounded for set valued maps. Denition 45. (Measurability) A set-valued mapping (multifunction) F : M! Cl(X ) is said to be measurable if it satises : fx2M :F (x)\O6= 0g2M for every open set OX: (2.2) It means, F is measurable if the inverse image of each open set is a measurable set. Denition 46. (Integrably Bounded) A measurable multifunctionF :M!Cl(X ) is said to be L p integrably bounded (p 1), if there exists h2L p (M;M;;R) such thatkak X h(!) for any a and ! with a2F (!). The denitions of 2 other important concepts - Integration and Conditional expectation for set valued functions and processes - require careful attention. 2.1 Integration First, we dene what we mean by Measurable selection which is going to be important in several theorems later. Denition 47. (Measurable Selection) Let F :M!Cl(X ) be measurable. A measurable map f : M!X which satises f(x)2 F (x) 8x2 M, is called a measurable selection of F . If f2L p (p 1), then it is called an L p integrable selection of F . We denote the set of all measurable selections of F asS(F ) and set of allL p integrable selections asS p (F ). Now, we are ready to dene integral of set valued maps which is called as Aumann Integral. 59 Denition 48. (Aumann Integral) The Aumann integral of F is the set of integrals of integrable selections of F : Z F d := Z fdjf2S p (F ) (2.3) We can easily prove the following properties. See for reference [22]. Properties : 1) If F is measurable, thenS(F )6=; 2) If F is L p integrably bounded, thenS p (F )6=; 3) (Convexity of the Integral) If F : M! Cl(R m ) and is nonatomic measure, then R F d is convex and if F is integrably bounded, then R F d is closed (and convex) LetP F = ( ;F;P;F) be a complete ltered probability space with ltrationF satisfying the usual hypothesis. Let G : [0;T ]!Cl(R m ) be measurable and integrably bounded. We have the following propositions. For proofs cf [22]. Proposition 49. The set valued mapping dened by (!) = R T 0 G(!;t)dt for !2 is measurable and : !K b c (R m ). Proposition 50. S() is a nonempty convex weakly compact subset of L( ;F;R m ). 2.2 Conditional Expectation LetG be a sub--algebra ofF. Given anF-measurable set-valued mapping : !Cl(R m ) with a nonempty set S() of all itsF-measurable and integrable selectors there existsG- measurable set-valued mapping E[jG] satisfying S(E[jG]) =cl L fE[jG] :2S()g (2.4) where cl L denotes the closure operation in L( ;G;R m ). We call E[jG] the multi-valued conditional expectation of relative toG. This conditional expectation has properties similar to those of the usual ones. For example, Z A E[jG]dP = Z A dP for everyA2G (2.5) where integrals are understood in Aumann's sense. 60 2.3 Measurable Selection Theorem Given a measurable, F-adapted and integrably bounded multivalued mapping G : [0;T ]! Cl(R m ) we denote by S F (G) a set of all measurable and F-adapted selectors for G. From Kuratowski and Ryll-Nardzewski measurable selection theorem it follows that for any given G, the set S F (G) is nonempty. It could be veried that S F (coG) is a nonempty convex and weakly compact subset of L([0;T ] ; F ;R m ) where coG represents convex hull of G and F is progressive -algebra. We present main theorem of this section. For proof, see [22] Theorem 9. LetG : [0;T ]!Cl(R m ) be a measurableF-adapted and integrably bounded multivalued mapping. Assume X = (X t ) 0tT is an m-dimensional cadlag process onP F such that E[jX T j]<1. Then X s 2E X t + Z t s G(;)djF s a.s. for every 0stT if and only if there is g2S F (coG) such that X t =E X T + Z T t g(;)djF s a.s. for every 0tT . 3 Backward Stochastic Dierential Inclusions We assume that we work on a complete probability spaceP F = ( ;F;P;F) with ltration F satisfying the usual hypothesis. Given measurable and uniformly integrable bounded set-valued mappings F : [0;T ] R d R m !Cl(R d ) andH : [0;T ]R m !Cl(R d ), a solution (Y;X) to Backward Stochastic Dierential Inclusion BSDI(F;H) means 8 > > < > > : Y s 2E Y t + Z t s F (;Y ;X )djF s a:s: Y T 2 Z T 0 H(t;X t )dt a:s: (3.1) for every 0stT . Moreover, Y and X should be cadlag processes. There are 2 notions of solutions to BSDIs : Strong Solutions and Weak Solutions. We dene them below. 61 Denition 51. (Strong Solution) IfP F and a cadlag processX are given thenY , satisfying above conditions with a ltration F X generated by X is said to be a strong solution to BSDI(F;H;X). Usually the driving process X is Brownian motion or a strong solution to a forward Stochastic Dierential Equation. Denition 52. (Weak Solution) For given multifunctions F and H and a probability mea- sure on Borel -algebra of D([0;T ];R m ) (Space of m-dimensional cadlag paths), we look for systems (P F ;Y;X) such that PX 1 = every F X - martingale is also a F-martingale a pair (Y;X) satises (2) and (3) onP F a.s. for every 0stT . Such systems are said to be weak solutions to BSDI(F;H;). 3.1 Existence and Compactness Results In this subsection, we present some existence and compactness results for BSDIs. For proofs refer to [22] LetA be equivalence class of all Aumann integrable set valued mappings on [0;T ] with metric d dened by d(A 1 ;A 2 ) = R T 0 H R d(A 1 (t);A 2 (t))dt for A 1 ;A 2 2A, where H R d is the Hausdor metric on the space Comp(R d ) of all nonempty compact subsets of R d . Let K : [0;T ]X! Comp(R d ) where (X;) is a given metric space. We say that K isA- continuous with respect to its second argument if a mapping X 3 x! K(;x)2A is continuous as a mapping from (X;) into (A;d). Theorem 10. (Existence) Let F : [0;T ]R d R m !Comp(R d ) and H : [0;T ]R m ! Comp(R d ) be measurable and uniformly integrable bounded and let be a probability measure on D(R m ). If F and H areA continuous, then BSDI(F;H;) possesses a weak solution. Theorem 11. (Weak Compactness) Let F : [0;T ]R d R m ! Comp(R d ) and H : [0;T ]R m ! Comp(R d ) be measurable, uniformly integrable bounded andA continuous. LetW(F;H; ) be set of all weak solutions to BSDI(F;H;) with 2 . If is weakly compact then the setW(F;H; ) is nonempty and weakly compact with respect to Meyer- Zheng Topology. 62 4 Existence of Optimal Weak Solutions In this section, we make connections between our problem and set-valued approach. For details and proofs, look at [22]. We would restrict equation (1.1) to the following form 8 > > < > > : Y u t =E Y T + Z T t f(s;Y u s ;X s ;u s )dsjF t for 0tT V 0 = sup u2U [0;T] '(Y u 0 ): (4.1) Here, f is a deterministic map f : [0;T ]R d R m U ! R d and X is a forward diusion. Instead of strong solutions, we would be looking for weak solutions. Let be weakly compact set of Probability measures on D([0;T ];R m ) andH : [0;T ]R m !Comp(R d ) be measurable, uniformly integrable bounded andA continuous. We consider a setW f (H; ) of all pairs (Y u ;X) satisfying BSDE (4.1) for some u2U, X such that PX 1 2 and Y T 2 R T 0 H(t;X t )dt. The setW f (H; ) is called an attainable set. It can be veried that W f (H; ) =W(F;H; ) where F (t;y;x) =ff(t;y;x;u) :u2Ug. If f is continuous and U is compact, then F : [0;T ]R d R m !Comp(R d ) andA continuous. Now, let J(Y u ) =E Y u ['(Y u 0 )]: (4.2) where ' is continuous. J is weakly compact with respect to Meyer Zheng Topology. From previous section, we know thatW(F;H; ) is nonempty and weakly compact with respect to Meyer-Zheng Topology. Hence there must exist an optimal trajectory Y u satisfying J(Y u ) = sup u2U J(Y u ) (4.3) 63 Chapter 8 Conclusions In this thesis, we transformed several time inconsistent problems as multi-dimensional con- trolled BSDEs optimization problems. We advocated that such controlled BSDEs are time inconsistent essentially due to the lack of comparison principle. We proposed 3 original methods on handling time inconsistency issues of controlled BSDE optimization problems. Here, we provide a brief summary of the 3 approaches. 1 Summary Each of the three approaches we proposed has their own pros and cons. It should be noted that in this thesis we focus mainly on the ideas, rather than the actual solvability of the resulting problems, which could be highly technical, and may call for some new developments in the respective areas. 1. The \Forward Utility" Approach. The main idea in this approach is to nd a certain random eld (t;!;y), (t;!;y)2 [0;T ] R d satisfying a weak Comparison Principle, so that V t := sup u2U (t;!;Y u t ) and (0;!;y) ='(y): (1.1) Moreover, we require that the pair (;V ) satisfy a certain DPP, and therefore the problem (1.1) is time-consistent (although the original problem ~ V t := sup u2U '(Y u t ), t2 [0;T ] is time-inconsistent!). This approach looks very promising and reasonable, but at this point we have only succeeded in nding the desired in the linear case. 2. The \Master Equation" Approach. This approach is motivated by the theory of mean-eld games. In fact, in light of the random eld in (1.1) and the backward nature of V , we consider a family of mappings (t;) :L 2 (F t )!R,t2 [0;T ], such thatV 0 = (T;). We shall argue that the familyf (t;) :t2 [0;T ]g is time-consistent in the sense that it is a solution to some HJB type equation, which we call the master equation. We note that, unlike the mean-eld game case, the function (t;) depends on the random variable , rather than the law of , thus it is the \lifted" version of those in mean-eld PDE (cf. e.g., 64 [4]), thus its Fr echet derivatives can be dened more directly. The main diculty of this approach, however, is that typically such will not have the desired smoothness, and it is a very challenging problem to propose appropriate notion of solution to the master equation under which is the unique solution. We shall leave this to future research. 3. Duality Approach. This approach is based on nding certain dynamic family of setsfD t :t2 [0;T ]g which satises a certain geometric DPP (see, e.g., [27]), and is therefore \time-consistent" in this sense. We shall then argue that V 0 = sup x2D 0 '(x): (1.2) We note that once the relationship (1.2) is established, then the problem is reduced to a nite dimensional optimization problem, which would be rather easy. However, this approach only produces the value of V 0 , but provides neither any information about the (dynamic) values V t , t> 0, nor any clue on the existence of optimal control. 2 Open Problems Although we addressed few issues in our work, however, there are several open problems left for future research. We list a few of them here. 1) We focused all our approaches on solving controlled BSDEs. However, many applica- tions (like Mean Variance optimization) fall in to the category of controlled FBSDEs. Hence, we would like to extend all our approaches to controlled FBSDEs 2) We worked with nite dimensional BSDEs. Some important applications like Opti- mal Stopping under distorted probability requires the model to be controlled innite dimensional BSDE. Hence, extending our work to innite dimensions is important. 3) We showed the existence of Forward Utility only in the linear case. We want to explore nonlinear cases where the existence of is possible. More ambitiously, we would like to characterize non-linearities under which we have the existence of . 4) We want to nd time consistent dynamics for the example of optimal stopping under probability distortion. (We want to give \Forward Utility" version to this problem.) 5) We would like to study the wellposedness of Master Equation. We would like to develop viscosity theory for non smooth solutions. 65 6) We want to explore the existence of optimal controls in general models using the approaches we developed for DPP - (Forward Utility, Master equation or Duality Approach). 66 Chapter 9 Appendix We now brie y review the so-called Functional Ito calculus. We refer the interested reader to [8] or [5] for the detailed theory. We consider the following canonical set-up: let := f!2C([0;T ];R d ) :! 0 = 0g, and B t (!) =! t , !2 , the coordinate (or canonical) process. We introduce the following pseudo-metric on and := [0;T ] : k!k t := sup 0st j! s j; d 1 ((t;!); (t 0 ;! 0 )) :=jtt 0 j 1 2 +k! t^ ! 0 t 0 ^ k T : (0.1) Let C 0 () be the set of processes u : ! R which is continuous under d 1 , and C 0 b () the subset of C 0 () which are bounded. We note that any u2 C 0 () is F-progressively measurable. Whenu is multidimensional, say taking values inR k , we denote it asC 0 (;R k ). Let P 0 be the Wiener measure, namely B is a Brownian motion under P 0 . We say a probability measureP on is a semi-martingale measure ifB is a semimartingale underP. LetS d denote the set of dd-symmetric matrices. Denition 53. Let u2 C 0 (). We say u2 C 1;2 () if there exist @ t u2 C 0 (;R);@ ! u2 C 0 (;R d );@ 2 !! u2 C 0 (;S d ) such that the following functional Ito formula holds: for any semimartingale measure P, du(t;!) =@ t udt +@ ! udB t + 1 2 @ 2 !! u :dhBi t ; P-a.s. (0.2) 67 Bibliography [1] Aubin, J.P. and Frankowska, H. (2008), Set-Valued Analysis, Birkhauser, 1st ed. 1990. 2nd printing 2008 Edition. [2] Bertsekas D. and Shreve S. (1978),Stochastic Optimal Control; the discrete-time case, Math. in Sci. and Eng., Academic Press. [3] Bjork, T. and Murgoci, A. (2010), A General Theory of Markovian Time Inconsistent Stochastic Control Problems, preprint, ssrn.com/abstract=1694759. [4] Cardaliaguet, P. (2013), Notes on Mean Field Games (from P.-L. Lions' lectures at Coll ege de France). https://www.ceremade.dauphine.fr/ cardalia/ [5] Cont, R. and Fournie, D. (2013), Functional It^ o Calculus and Stochastic Integral Rep- resentation of Martingales, Annals of Probability, 41, 109-133 [6] Cvitani c, J. and Zhang, J. (2012), Contract Theory in Continuous Time Models, Springer Finance. [7] David M. Kreps (2013), Microeconomic Foundations I : Choice and Competitive Markets, Princeton University Press. [8] Dupire, B. (2009), Functional It^ o calculus, preprint, papers.ssrn.com. [9] Ekeland, I. and Lazrak, A. (2010), The golden rule when preferences are time inconsis- tent, Math. Financ. Econ., 4, 29-55. [10] Ekren, I., Touzi, N., and Zhang, J., Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I, Annals of Probability, to appear. [11] Ekren, I., Touzi, N., and Zhang, J., Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part II, Annals of Probability, to appear. [12] Ekren, I. and Zhang, J., Wellposedness of Degenerate Fully Nonlinear PPDEs, preprint. [13] El Karoui, N. and Mrad, M. (2010), An exact connection between two solvable SDEs and a nonlinear utility stochastic PDEs, preprint, arXiv:1004.5191. 68 [14] Feinstein, Z. and Rudlo, B. (2013), Time consistency of dynamic risk measures in markets with transaction costs, Quantitative Finance, 13, 1473-1489. [15] Feinstein, Z. and Rudlo, B. (2016), Time consistency for scalar multivariate risk measures, working paper. [16] Frederick, S., Loewenstein, G. and O'Donoghue, T. 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Creator Karnam, Chandrasekhar (author)

Core Title Dynamic approaches for some time inconsistent problems

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 04/22/2016

Defense Date 03/23/2016

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