Stochastic Differential Games Without Isaacs Condition

Differential games can be regarded as a type of optimal control problems with two (multiple) controls, and the classical optimal control problems can also be seen as differ-ential games with one single player. Isaacs [53] was the first to investigate games whose dynamics were given by the differential equations, namely, differential games. Since then, many authors studied the differential games and their applications in particular in finance and social sciences. Fleming and Souganidis [46] considered the so-called two-player zero-sum stochastic differential games, and they proved the existence of the value under the well-known Isaacs condition. Their work extended the result of Evans and Souganidis [40] to the stochastic case. With the development of the theory of backward stochastic differential equations, many researchers investigated the differential games by using backward stochastic differential equations. Hamadene, Lepeltier [49] studied the zero-sum stochastic differential games by using the results of backward stochastic differential equations. Hamadene, Lepeltier, Peng [51] constructed the Nash equilibri-um points of the nonzero-sum stochastic differential games by using the solutions of the related backward stochastic differential equations. Buckdahn, Cardaliaguet. Rainer [9] considered nonzero-sum stochastic differential games and obtained the existence of the equivalent Nash equilibrium payoffs. Cardaliaguet [26] was the first to study the differential games with asymmetric information and proved the existence of the value.The existence problem of the value of the zero-sum stochastic differential games without Isaacs condition is an open problem. The first to study deterministic differential games without Isaacs condition were Krasovskii and Subbotin [60]. using the positional strategies they proved the existence of the value. Buckdahn, Li and Quincampoix [22] studied zero-sum differential games without assuming Isaacs condition, and they proved the existence of the value by using non-anticipative mixed strategies with delay. In [23], they generalized their work [22] to the stochastic case.In our paper, without assuming Isaacs condition, we mainly study two-player zero-sum differential games and also, two-player nonzero-sum differential games in both cases. the deterministic and the stochastic one. Moreover, we also consider a new type of reflected mean-field backward stochastic differential equations (reflected MFBSDEs, for short), namely the controlled reflected MFBSDEs coupled with the value function.Let us introduce the content and structure of this thesis.In Chapter 1 we give the introduction of our paper.In Chapter 2 we consider two-player zero-sum differential games without Isaacs condition, where the cost functional is a terminal pay-off with asymmetric information. To overcome the absence of Isaacs condition and hide the private information of each player, a randomization of the non-anticipative strategies with delay (NAD strategies, for short) of the both players is considered. It differs from that in Buckdahn, Quincampoix, Rainer and Xu [25]. Unlike in [25], our definition of NAD strategies for a game over the time interval [t.T] (0< t< T), depends on the events occurring before the game starts and it guarantees that a randomized strategy along a partition Ï€ of [0,T] remains a randomized NAD strategy with respect to any finer partition π’(Ï€ (?) π’). This allows to study the limit behavior of upper and lower value functions for games in which the both players can use different partitions depending on their individual choice. We investigate the properties of the upper and lower value functions, the related sub-dynamic programming principle and the associated Hamilton-Jacobi-Isaacs equations with the help of Fenchel transformation, and we prove the existence of the value.The novelty of this chapter:We extend the definition of the non-anticipative strategy with delay in Buckdahn, Quincampoix, Rainer and Xu [25], and prove the existence of the value of our differential games without Isaacs condition and with asymmetric information. Then we give the characterization of the value which is the key for numerical approaches.In Chapter 3 using the same framework of Chapter 2 but now with different pay-off functionals for both players, we study two-player nonzero-sum differential games with symmetric information. Our main result here proves, the existence of the Nash equilibrium payoffs. Our results extend those by Buckdahn, Cardaliaguet and Rainer [9] to the case without assuming Isaacs condition.The novelty of this chapter:We are the first to consider the nonzero-sum differ-ential games without Isaacs condition. By the characterization of the equivalent Nash equilibrium payoffs, we prove the existence of the Nash equilibrium payoffs.The Chapters 2 and 3 of the present paper are based on:J. Li, W. Li. Zero-sum and nonzero-sum differential games without Isaacs condition. Submitted, http://arxiv.org/abs/1507.04989.In Chapter 4 we generalize the results of Chapter 3 to stochastic case, that is we investigate the existence of the Nash equilibrium payoffs for nonzero-sum stochastic differential games without assuming Isaacs condition. Along a partition Ï€ of the time interval [0,T] we choose a suitable random non-anticipative strategy with delay to study our nonzero-sum stochastic differential game. We prove for the corresponding both zero-sum stochastic differential games without Isaacs condition the value functions exist. With the help of these value functions we give a characterization of the Nash equilibrium payoff. This characterization allows us to prove the existence of the Nash equilibrium payoffs.The novelty of this chapter:We are the first to consider the nonzero-sum stochastic differential games without Isaacs condition. By the characterization of the equivalent Nash equilibrium payoffs, we prove the existence of the Nash equilibrium payoffs.The Chapter 4 of the present paper is based on:J. Li, W. Li. Nash equilibrium payoffs for nonzero-sum stochastic differential games without Isaacs condition. Submitted.In Chapter 5 we consider a new type of reflected MFBSDEs, namely, controlled reflected MFBSDEs involving their value function. The existence and the uniqueness of the solution of such equation are proved by using an approximation method. We also adapt this method to give a comparison theorem for our reflected MFBSDEs. The related dynamic programming principle is obtained by extending the approach of stochastic backward semigroups introduced by Peng [104] in 1997. Then, we show that the value function which our reflected MFBSDE is coupled with is the unique viscosity solution of the related nonlocal parabolic partial differential equation with obstacle.The novelty of this chapter:We generalize the results of Hao and Li [50] to the re-flected case. We prove that the reflected MFBSDE coupled with the value function exists a unique solution, and provides a probabilistic interpretation for an obstacle problem of the associated parabolic partial differential equation.The Chapter 5 of the present paper is based on:J. Li, W. Li. Controlled reflected mean-field backward stochastic differential equa-tions coupled with value function and related PDEs. Mathematical Control and Related Fields,5 (3),501-516,2015.This paper include five chapters, we now give an outline of the structure and the main conclusion of this dissertation.Chapter 1 Introduction:Chapter 2 Zero-sum differential game without Isaacs condition and with asym-metric information;Chapter 3 Nash equilibrium payoffs for nonzero-sum differential game without Isaacs condition;Chapter 4 Nash equilibrium payoffs for nonzero-sum stochastic differential game without Isaacs condition;Chapter 5 Controlled reflected MFBSDE coupled with value function and related PDEs. Chapter 2:We prove that the upper value function WÏ€ and the lower value function VÏ€ uniformly converge to the same function U when the mesh of the partition Ï€ tends to 0 and the value function U is the unique dual viscosity solution of some Hamilton-Jacobi-Isaacs equation. On the other hand, we give the characterization of the value function U, and prove that the upper value function W and the lower value function V are equal when the Isaacs condition holds, namely, we have W= U= VFor any given t ∈ [0,T],× ∈ Rn, we consider the following dynamicsFor (p,q) ∈△(I) ×△(J), (t, x) ∈ [0,T] × Rn, Ï€={0=t0<11<…<tN= T} and t ∈[tk-1,tk), we define the following cost functional We now introduce the following upper value functions and lower value functions whereIn order to get the existence of the value, we study the upper and lower value func-tions WÏ€ and VÏ€ along the partition Ï€. With the help of the Girsanov transformation introduced in Buckdahn, Li [16], we investigate the upper and lower value functions (WÏ€,VÏ€) by the functions (W1Ï€, V1Ï€). Combined with the Fenchel transformation intro-duced in Cardaliaguet [26] or Buckdahn, Quincampoix, Rainer, Xu [25], we prove that (W1Ï€, V1Ï€) converge uniformly to the same function, namely the value function.Theorem 2.2.1 For any (t,x,p, q) ∈ [0,T] ×Rnâ–³(I)×△(J), we have VÏ€(t,x,p,q)=V1Ï€ (t,x,p,q), WÏ€(t,x,p,q)=W1Ï€(t,x,p,q), whereLemma 2.2.2 The functions W1Ï€ and V1Ï€ are Lipschitz continuous in (t,x,p,q), uni-formly with respect to Ï€.Lemma 2.2.3 For any (t,Ï€x) ∈ [0,T] ×Rn, the functions W1∠(t,x,p,q) and V1∠(t,x,p,q) are convex in p ∈ â–³(I) and concave in q ∈△(J).Lemma 2.2.4 For all(t,x,p,q) ∈ [0,T] ×Rn×RI×△(J). we haveLemma 2.2.5 For any partition Ï€ of the interval [0,T], the convex conjugate function V1Ï€*(t,x,p,q) is Lipschitz continuous with respect to(t,x,p, q),and the concave conjugate function W1Ï€#(t,x.p, q) is Lipschitz continuous with respect to (t.x.p,q). The Lipschitz constants are independent of Ï€.Lemma 2.2.6 (sub-DPP) For all (t,x,p,q) ∈ [tk-1,tk)×Rn×RI×△(J), and for all l (k≤l≤TV), we haveLemma 2.2.7 There exists a subsequence of (Ï€n)n≥1, still denoted by (Ï€n)n≥1, and two functions V:[0,T]×Rn×RI×△(J)â†'R and W:[0,T]×Rn×△(I)×RJâ†'R such that (VÏ€n*,W1Ï€Wn#)â†'(V,W) uniformly on compacts in [0,T] ×Rn×△(I)×△(J)×RI×RJ. Lemma 2.2.8 For all (p,q) ∈ RI×△(J), the limit function V(t,x,p,q) is a viscosity subsolution of the HJI equation (2.2.36).Lemma 2.2.9 For any (t,x,p,q) ∈ [0,T] ×Rn×△(I)×RJ, and for all l (k≤l≤n), we have and W (Recall Lemma 2.2.7) is a viscosity supersolution of the HJI equation (2.2.36).Theorem 2.2.2 The functions (V1Ï€n) and (W1Ï€n) converge uniformly on compacts to the same Lipschitz function U when the mesh of the partition Ï€n tends to 0. Moreover, the function U is the unique dual viscosity solution of the HJI equation (2.2.51).Theorem 2.2.3 The functions (VÏ€n) and (WÏ€n) converge uniformly on compacts to the same Lipschitz function U when the mesh of the partition Ï€n tends to 0. Moreover, the function U is the unique dual viscosity solution of the HJI equation (2.2.51).In order to study the property of the upper value function W and lower value function V, we need to introduce the following functionsUsing the method introduced in Cardaliaguet [26] or Buckdahn, Quincampoix, Rain-er. Xu [25] again, we obtain the following main results.Theorme 2.3.1 Suppose condition (2.3.6) holds. Then, for all sequences of partitions (Ï€n) with|Ï€n|â†'0, the sequences (VÏ€n) and (WÏ€n) converge uniformly on compacts to the same Lipschitz continuous function U. Moreover, the function U is the unique dual viscosity solution of the HJI equation (2.2.51).Theorem 2.3.2 Suppose condition (2.3.7) holds. Then, for all sequences of partitions (Ï€n) with|Ï€n|â†'0, the sequences (VÏ€n) and(WÏ€n) converge uniformly on compacts to the same Lipschitz continuous function U. Moreover, the function U is the unique dual viscosity solution of the HJI equation (2.2.51).Theorem 2.3.3 (Characterization of the value) Under Isaacs condition, it holds W(t,x,p,q)= U(t,x.p,q)= V(t,x,p,q), for all (t,x,p,q) ∈ [0,T]×Rn×△(I)×△(J).Chapter 3:We mainly study the existence of the Nash equilibrium payoff for the nonzero-sum differential game without Isaacs condition. We first give the characterization of the Nash equilibrium payoff, then prove the existence of the Nash equilibrium payoff.Under the framework of Chapter 2. we study the nonzero-sum differential game with the symmetric information (i.e. I=J=1). The payoffs are the following cost functionals The aim of Player 1 is to maximize J1(t,x,u,v), while player 2 wants to maximize J2(t,x,u,v). Using the approach in Buckdahn, Cardaliaguet, Rainer [9] and the following value functions and we prove the existence of the Nash equilibrium payoff (NEP, for short) for the nonzero-sum differential game without Isaacs condition.Lemma 3.2.1α)Let(t,x)∈[0,T]×Rn and ∈>0.Then for any partition Ï€={0= t0<t1<…<tN=T)with |Ï€|<δε(δε>0 small enough)t=tk-1,and for any given u’ ∈ut,TÏ€,1,there exist strategies αi ∈A1Ï€(t,T),i=k,…,N,such that for all u ∈Vt,TÏ€,1, b)Let(t,x)∈[0,T]×Rn and ε>0.Then for any partition Ï€={0=t0<t1<…< tN=T}with |Ï€|<δε(δε>0 small enough)t=tk-1,and for any given u’ ∈ut,TÏ€,1,there exist strategies αi ∈A1Ï€(t,T),i=k,…,N,such that for all v ∈Vt,TÏ€,1,Theorem 3.2.1(Characterization of NEP)A couple(e1,e2) ∈R2 is a Nash equilibrium payoff at the position(t,x)if and only if,for all ε>0,there exists δε>0 satisfying that for any partition Ï€={0=t0<t1<…<tN=T)with |Ï€|<δε and t=tk-1,there exists(uε,vε) ∈Ut,TÏ€.1×Vt,TÏ€,1 that,for i=k,…,N and m=1,2,respectively, andProposition 3.2.1 For any ∈>0,there exists δε>0 small enough satisfying that,for ally partition Ï€={0=t0<t1<…<tN=T} with |Ï€|<δε and t=tk-1,there exists a pair(uε,vε) ∈ut,TÏ€.1×Vt,TÏ€,1 such that,for all k≤l≤N,andmn=1,2, where X=Xt,x,uε,vε.Theorem 3.2.2(Existence of NEP)For any initial position(t,x)∈[0,T]×Rn,there exists a Nash equilibrium payoff at the position(t,x).Chapter 4:We extend the result of Chapter 3 to the stochastic case,i.e. nonzero-sum stochastic differential game. We first consider the associated zero-sum stochastic differential game and show the existence of the value, then we prove the existence of the Nash equilibrium payoff with the help of its characterization. Our dynamic is the following doubly controlled stochastic differential equation Zero-sum stochastic differential game:For (tmx) ∈ [0, T]×Rn, Ï€={0=t0<t1<<tN=T} and t ∈ [tk-1,tk), (α,α) ∈ At,Tπ×Bt,TÏ€, we define the following cost functional Let us introduce the following upper and lower value functions along the partition Ï€:For the zero-sum stochastic differential game without Isaacs condition, we mainly study the properties of the upper and lower value functions UÏ€ and WÏ€. We prove that, the upper and lower value functions UÏ€ and WÏ€ have deterministic versions and satisfy the dynamic programming principle with respect to the points of the partition. Moreover, when the mesh of the partition Ï€ tends to 0, the functions WÏ€ and UÏ€ converge uniformly to the same function which is the unique viscosity solution of some Hamilton-Jacobi-Bellman-Isaacs (HJBI, for short) equation.Theorem 4.2.1 For all partitions Ï€ of the time interval [0, T], (t,x) ∈ [0,T]×Rn, we haveTheorem 4.2.2 Let Ï€={0=t0<t1<…<tN=T},t ∈[tk-1,tk),x ∈ Rn. Then for all k≤l≤N, we have, P-a.s.,Lemma 4.2.1 For all(t,x)∈[0,T]×Rn,it holdsProposition 4.2.1 For all partitions Ï€ of the time interval[0,T],there is some constant C>0 such that for all t,r ∈[0,T]x,y ∈Rn,we haveTheorem 4.2.3 There exists a bounded continuous function V:[0,T]×Rnâ†'R such that for all partitions Ï€n of interval[0,T]satisfying Ï€nâ†'0,as nâ†'∞,the upper and lower value functions(UÏ€n,WÏ€n)uniformly converge to(VmV)on compacts in[0,T]×Rn. Moreover,V is the unique viscosity solution of the HJBI equation(4.2.52).Nonzero-sum stochastic differential game:For any given(t,x) ∈[0,T]×Rn and partition Ï€ ={0=t0<t1<…<tN=T)with t ∈[tk-1,fk),we defineBoth Players want to maximize their cost functionals.In order to get the existence of the Nash equliibrium payoff,we introduce the value functions W1(t,x)and W2(t,x) associated with g1 and 92:respectively,Lemma 4.3.2 a)Let(t,x)∈[0,T]×Rn and ∈>0.Then for any partition Ï€={0= t0<t1<…<tN=T}with |Ï€|<δε(δε>0 small enough)t=tk-1,and for any given u’ ∈ut,TÏ€, there exist NAD strategies αi ∈At,TÏ€,i=k-1,…,N,such that for all v ∈Vt,TÏ€, b)Let(t,x)∈[0,T]×Rn and ε>0.Then for any partition Ï€={0=t0<t1<…< tN=T)with |Ï€|<δε(δε>0 small enough)t=tk-1,and for any given u’ ∈ut,TÏ€,there exist NAD strategies αi ∈At,TÏ€,i=k-1,…,N,such that for all v ∈Vt,TÏ€,Theorem 4.3.1(Characterization of NEP)A couple(el,e2) ∈R2 is a Nash equilibrium payoff at the position(t,x)if and only if,for any ε>0,there exists δε>0 small enough satisfying that for any partition Ï€={0=t0<t1<…<tN=T}with |Ï€|<δε and t=tk-1,there exists(uε,uε) ∈ut,Tπ×Vt,TÏ€ such that,for i=k-l,…,N and m=1,2, respectively, and Proposition 4.3.1 For any ε>0,there exists δεE>0 small enough satisfying that,for any partition Ï€={0=t0<t1<…<tN=T}with |Ï€|<δε and t=tk1,there exists a pair(tε,uε) ∈ut,Tπ×Vt,TÏ€ such that,for all k-1≤i≤l≤N,and m=1,2, where X=Xt,x,uεmuε.Theorem 4.3.2(Existence of NEP)For any initial position(t,x)∈[0,T]×Rn,there exists a Nash equilibrium payoff for the nonzero-sum stochastic differential games.Chapter 5:We consider a new type of reflected MFBSDEs,namely,con-trolled reflected MFB SDEs involving their value function.The existence and the uniqueness of the solution of such equation are proved by using an approx-imation method. We also adapt this method to give a comparison theorem for our reflected MFBSDEs.The related dynamic programming principle is obtained by extending the approach of stochastic backward semigroups in-troduced by Peng[104]in 1997.Then,we show that the value function which our reflected MFBSDE is coupled with is the unique viscosity solution of the related nonlocal parabolic partial differential equation with obstacle.McKean-Vlasov SDE:Given (x0,v) ∈ RnxV0,T, for all t ∈ [0,T], ζ∈L2(Ω,Ft,P and v(·) ∈ Vt,T, we consider the following SDE Under the assumption (H5.2.1), the solution of this equation is unique, denoted by Xt,ζ;vWe consider the following reflected MFBSDE, namely reflected MFBS-DE coupled with the value function:We use an iterative approach to study the existence of the solution of equation (5.2.3). Putting Yt,x;v,0=0, and letting m≥1, the iterative equation is as follows. For (t,x) ∈ [0,T] ×Rn,v ∈Vt,T, we considerLemma 5.2.1 For all m≥1,(5.2.4)admits a unique solution (Yt,x;vmm,Zt,x;v,m,Kt,v;v,m) ∈SF2(t,T;R)×HF2(t,T;Rd)×AF2,c(t,T;R),(t,x)∈[0,T]×Rn,u ∈Vt,T.Moreover, Wm:Ω×[0,T]×Rnâ†'R is a measurable,a priori random function such that(i)Wm(t,x)is Ft-measurable,(t,x) ∈[0,T]×Rn;(ii)There exists a constant C independent of m:such that,for all t ∈[0,T], x.x ∈Rn.P-a.s..Theorem 5.2.1(Existence)For all(t,x) ∈[0,T]×Rn,v ∈Vt,T,there exists a triplet of processes(Yt,x;v,Zt,x;v,Kt,v;v) ∈ SF2(t,T;R)×HF2(t,T;Rd)×AF2,c(t,T;R), such thatYt,x;vmm,Zt,x;v,m,Kt,v;v,m)m≥1 converges to(Yt,x;v,Zt,x;v,Kt,v;v)in SF2(t,T;R)× HF2(t,T;Rd)×AF2,c(t,T;R),and Wm(t,x),m≥1,converges to W(t,x)= in L2.Moreover,(Yt,x;v,Zt,x;v,Kt,v;v,W),v ∈Vt,T,(t,x)∈[0,T]×Rn,solves reflected MFBSDE (5.2.3).Furthermore,there exists a constant C>0 such that,P-a.s.,for all t ∈[0,T],x,x ∈Rn,We haveTheorem 5.2.2(Uniqueness)Under the assumptions(H5,2.1)and(H5.2.2),the solu-tion of reflected MFBSDE(5.2.3)coupled with its value fumction is unique.Theorem 5.2.3(Comparison Theorem)We assume the drivers fi=fi(t,x’x,x,y’,y,z) and the obstacles hii(t,x’,x)satisfy(H5.2.2)and(H5.2.3)with the terminal value εi ∈ L2(Ω,FT,P),i=1,2.Let(Yi,t,x;v,Zi,t,x;v,Ki,t,x;v,Wi),u ∈Vt,T,t ∈[0,T],x ∈Rn,i= 1,2,be the unique solutions of reflected MFBSDE(5.2.3)with data(fi,ζi,hii),i=1,2, respectively.Moreover,let ζ1≥ζ2,h1≥h2,and f1≥f2.Then Ys1,t,x;v≥Ys2,t,x;v,P-a.s., s ∈[t,T],(t,x,v)∈[0,T]×Rn×Vt,T,and W1(t,x)≥W2(t,x),(t,x)∈[0,T]×Rn.Theorem 5.3.1(DPP)Under the assumptions(H5.2.1)and(H5.2.2),for all t ∈[0,T), z ∈Rn,and 0≤δ<T-t,the value function W(t,x)obeys the following DPP:Theorem 5.4.1(i)(Existence)Under the assumptions(H5.2.1)and(H5.2.2),the value function W ∈Cp([0,T]×Rn)given by the reflected MFBSDE(5.2.3)in Theorem 5.2.1 is a viscosity solution of PDE(5.4.1). (â…±) (Uniqueness) The value function W in (â…°) is the unique viscosity solution of PDE (5.4.1) in the space (?).