# A BSDE Approach to Stochastic Differential Games with Regime Switching

Authors: J. Y. Li; M. N. Tang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we study a two-player zero-sum stochastic differential game with regime switching in the framework of forward-backward stochastic differential equations on a finite time horizon. By means of backward stochastic differential equation methods, in particular that of the notion from stochastic backward semigroups, we prove a dynamic programming principle for both the upper and the lower value functions of the game. Based on the dynamic programming principle, the upper and the lower value functions are shown to be the unique viscosity solutions of the associated upper and lower Hamilton–Jacobi–Bellman–Isaacs equations.


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## Body

## 1. Introduction

The differential game is concerned with the problem that multiple players make decisions, according to their own advantages and trade-off with other partners in a dynamic system. Stochastic differential games (SDGs) have been well studied. Recently, Lv [1] studied the two-player zero-sum SDGs in a regime switching model with an infinite horizon. Compared with the traditional diffusion model, the regime switching model has two obvious advantages. First, the underlying Markov chain can be used to model discrete events with larger long-term system impact. For instance, in financial markets, it is easy to capture market trend by using finite state Markov chain. However, it is difficult to incorporate this dynamic into pure diffusion model. Second, when conducting numerical experiments, regime switching models require very limited data input. In recent years, due to the capacity for characterizing all kinds of random events and the tractability, regime switching models have attracted extensive attention [1–3]. In this paper, we introduce a new method, which is different from the method in [1]. We investigate two-player zero-sum SDGs with regime switching on a finite time horizon by using the backward stochastic differential equation (BSDE) methods.Pardoux and Peng [4] first introduced the nonlinear BSDEs in 1990. The theory of BSDE was originally developed by Peng [5] for stochastic control theory. And later Hamadène and Lepeltier [6] and Hamadène et al. [7] introduced this theory to SDGs. Buckdahn and Li [8] studied a recursive SDG problem and interpreted the relationship between the controlled system and the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation. The theory of BSDEs has been well studied and applied to many fields, such as stochastic control, SDGs, mathematical finance, and partial differential equation theory (see [5–7, 9–11] for details). The readers interested in other topics about game theory are referred to [12–15].In this paper, letΩ,ℱ,ℙ be a fixed probability space on which a d-dimensional Brownian motion Bss∈0,T and a Markov chain θss∈0,T are defined on some sample space Ω, ℱ is the completed Borel σ-algebra over Ω, and P is the Wiener measure. Here, we assume that ℱ=ℱss∈0,T, where ℱs≜ℱsθ∨ℱsB. Let ℱsBs∈0,T denote the filtration generated by Brownian motion Bss∈0,T. And ℱsθs∈0,T denote the filtration generated by the Markov chain θss∈0,T. Assume that B⋅ and θ⋅ are independent. The Markov chain θss∈0,T takes values in a finite state space ℳ=1,…,m and is observable. And the generator Q=qiki,k∈ℳ∈Rm×m of the Markov chain θss∈0,T is given by(1)Pθs+δ=k|θs=i,s∈0,T=qikδ+oδ,ifk≠i,1+qiiδ+oδ,ifk=i,where qik is the transition rate from market regime i to k, qii=−∑k≠iqik<0, and qik≥0, for every i,k∈ℳ×ℳ.We will investigate a two-player zero-sum SDG with regime switching in the framework of BSDE on a finite time horizon. The dynamics of the SDG are described by the following functional stochastic differential equation (SDE): fort∈0,T,(2)dXst,x,i;u,v=bs,Xst,x,i;u,v,θst,i,us,vsds+σs,Xst,x,i;u,v,θst,i,us,vsdBs,s∈t,T,Xtt,x,i;u,v=x,θtt,i=i,where T>0 is a fixed finite time horizon, t,x,i∈0,T×Rn×ℳ is regarded as the initial state, and Pθtt,i=i=1. Xt is the value of X at time t, and u,v=us,vss∈t,T are the pair of ℱt-adapted processes, take their values in some compact metric spaces U and V, and are called admissible controls of the two players I and II, respectively. Precise assumptions on the coefficients b and σ are given in the next section.The cost functional is introduced by BSDE:(3)Yst,x,i;u,v=ΦXTt,x,i;u,v,θTt,i+∫sTfr,Xrt,x,i;u,v,Yrt,x,i;u,v,Zrt,x,i;u,v,θrt,i,ur,vrdr−∫sTZrt,x,i,u,vdBr,s∈t,T,where Xt,x,i;u,v and θt,i are introduced in (2). The above BSDE has a unique solution Yst,x,i;u,v,Zst,x,i;u,vs∈t,T. And for given control processes u∈Ut,T and v∈Vt,T, we introduce the associated cost functional:(4)Jt,x,i;u,v≔EYtt,x,i;u,v|ℱt,t,x,i∈0,T×Rn×ℳ,where Yt,x,i;u,v is defined by BSDE (3). In the game, player I aims to maximize (4) and contrarily player II aims to minimize (4). We define the lower and the upper value functions W and U, respectively:(5)Wt,x,i≔essinfβ∈ℬt,Tesssupu∈Ut,TJt,x,i;u,βu,(6)Ut,x,i≔esssupα∈At,Tessinfv∈Vt,TJt,x,i;αv,v.Precise definitions ofα and β are given in the next section. In the case W=U we say that the game admits a value. The main objective of this paper is to show that W and U are, respectively, the unique viscosity solutions of the following lower and upper HJBI equations, and both are systems consisting of m coupled equations:(7)∂∂tWt,x,i+H−t,x,i,W,DW,D2W+∑k≠iqikWt,x,k−Wt,x,i=0,WT,x,i=Φx,i,(8)∂∂tUt,x,i+H+t,x,i,U,DU,D2U+∑k≠iqikUt,x,k−Ut,x,i=0UT,x,i=Φx,i,associated with(9)H−t,x,i,r,p,A,u,v=supu∈Uinfv∈VHt,x,i,r,p,A,u,v,H+t,x,i,r,p,A,u,v=infv∈Vsupu∈UHt,x,i,r,p,A,u,v,where H is defined as(10)Ht,x,i,p,A,u,v=12TrσσTt,x,i,u,v⋅A+bt,x,i,u,v,p+ft,x,i,r,σTt,x,i,u,vp,u,v,for t,x,i,r,p,A,u,v∈0,T×Rn×ℳ×R×Rn×Rn×n×U×V⟶R. If Isaacs’ condition holds, i.e., H−=H+, then (5) and (6) coincide, and the uniqueness of viscosity solution implies W=U, that is, the game admits a value.The paper is organized as follows. In Section2, we introduce some notations and preliminaries, which will be needed in what follows. In Section 3, we introduce the dynamic programming principle. In Section 4, based on the dynamic programming principle, we investigate that the upper and the lower value functions are the unique viscosity solutions of the associated upper and lower HJBI equations.


## 2. Preliminaries

Let us introduce the following spaces, which will be needed in what follows.(11)L2Ω,ℱT,P;Rn;=ξ:Ω⟶RnisℱT−measurable variable such thatEξ2<+∞,S20,T;Rn≔ψtt∈0,TRn−valuedℱTθ∨ℱB−adapted continous process:Esupt∈0,Tψt2<+∞,ℋ20,T;Rn≔ψtt∈0,TRn−valuedℱTθ∨ℱB−adapted process:E∫0Tψt2dt<+∞.We consider the BSDE with dataf,ξ:(12)Yt=ξ+∫tTfs,Ys,Zsds−∫tTZsdBs,t∈0,T.Heref:Ω×0,T×R×Rd⟶R is such that, for any y,z∈R×Rd, f⋅,y,z is ℱTθ∨ℱB-progressively measurable. We make the following assumptions:(A1) There exists a positive constantC such that for all t,yi,zi∈0,T×R×Rd,i=1,2,(13)ft,y1,z1−ft,y2,z2≤Cy1−y2+z1−z2,a.s.(A2)f⋅,0,0∈ℋ20,T;R.Lemma 1.
Let assumptions (A1) and (A2) hold; then, for any random variableξ∈L2Ω,ℱT,P, BSDE (12) has a unique solution:(14)Y,Z∈S20,T;R×ℋ20,T;Rd.We give the comparison theorem for solutions of BSDEs.Lemma 2.
(comparison theorem). Letξ1,ξ2∈L2Ω,ℱT,P;R and f1 and f2 satisfy (A1) and (A2). We denote by Y1,Z1 and Y2,Z2 the solutions of BSDEs with data f1,ξ1 and f2,ξ2, respectively, and we suppose that(i)
ξ1≤ξ2,P−a.s.(ii)
f1t,Yt2,Zt2≤f2t,Yt2,Zt2,dtdP−a.e.Then, we haveYt1≤Yt2,a.s., for all t∈0,T. Moreover, if Pξ1<ξ2>0, then PYt1<Yt2>0,t∈0,T, and in particular, Y01<Y02.With the notations in the above lemma, we assume that, for somef:0,T×R×Rd⟶R satisfying (A1) and (A2), the drivers fi,i=1,2 have the following form:(15)f1s,y,z=fs,y,z+φ1s,f2s,y,z=fs,y,z+φ2s.Then, we have the following lemma.Lemma 3.
The difference of the solutionsY1,Z1 and Y2,Z2 of BSDE (12) with the data ξ1,f1 and ξ2,f2, respectively, satisfies the following estimate:(16)Yt1−Yt2+12E∫tTeβt−sYs1−Ys22+Zs1−Zs22ds|ℱt≤EeβT−tξ1−ξ22|ℱt+E∫tTeβt−sφ1s−φ2s2ds|ℱt,where β=161+C2 and C is the Lipschitz constant in (A1).For the proof of the above two lemmas, the readers can refer to [11, 16].We now consider the assumptions on the coefficientsb and σ. The coefficients b:0,T×Rn×ℳ×U×V⟶Rn and σ:0,T×Rn×ℳ×U×V⟶Rn×d are two given functions. The mappings b and σ satisfy the following conditions:(A3)(i)
For every fixedx∈Rn,i∈ℳ, b⋅,x,i,⋅,⋅ and σ⋅,x,i,⋅,⋅ are continuous with respect to t,u,v.(ii)
For anyx,x′∈Rn,i∈ℳ, u∈U and v∈V, there exists a positive constant C such that(17)bt,x,i,u,v−bt,x′,i,u,v+σt,x,i,u,v−σt,x′,i,u,v≤Cx−x′.From (A3), we can get the global linear growth conditions ofb and σ, i.e., the existence of some C>0 such that, for all t∈0,T, u∈U, v∈V, x∈Rn, i∈ℳ,(18)bt,x,i,u,v+σt,x,i,u,v≤C1+x.Suppose the above assumptions hold; for anyu∈U and v∈V, control system (2) has a unique solution Xst,x,i;u,v,s∈t,T. And we have the following estimates.Lemma 4.
Under the assumptions of the mappingsb and σ, there exists a positive constant C such that, for any x,x′∈Rn,i∈ℳ, and u∈U,v∈V,(19)Esups∈t,TXst,x,i;u,v2|ℱt≤C1+x2,P−a.s.,Esups∈t,TXst,x,i;u,v−Xst,x′,i;u,v2|ℱt≤Cx−x′2,P−a.s.Suppose that the two functionsf:0,T×Rn×R×Rd×Rm×m×U×V⟶R and the terminal cost Φ:Rn⟶R satisfy the following conditions:(A4)(i)
For any fixedx,y,z,i∈Rn×R×Rd×ℳ, f⋅,x,y,z,i,⋅,⋅ is continuous with respect to t,u,v.(ii)
There existsC>0 such that, for all t∈0,T, x,x′∈Rn, y,y′∈R, z,z′∈Rd, i∈ℳ, u∈U and v∈V,(20)ft,x,y,z,i,u,v−ft,x′,y′,z′,i,u,v≤Cx−x′+y−y′+z−z′.(iii)
There is a constantC>0 such that, for all x,x′∈Rn, i∈ℳ,(21)Φx,i−Φx′,i≤Cx−x′.Under the above conditions, (3) has a unique solution Yst,x,i;u,v,Zst,x,i;u,vs∈t,T. And we have the following estimates.Lemma 5.
For allt∈0,T, i∈ℳ, u∈U, and v∈V, there exists a constant C>0 such that,(22)Ytt,x,i;u,v≤C1+x,P−a.s.,Ytt,x,i;u,v−Ytt,x′,i;u,v≤Cx−x′,P−a.s.
For the proof of this lemma, the readers can refer to [17].Now, we introduce the admissible controls and admissible strategies. Lett1, t2 be two deterministic times, and 0≤t≤t1≤t2≤T.Definition 1.
An admissible control processu=ur,r∈t1,t2 (resp., v=vr,r∈t1,t2) for player I (resp., player II) on t1,t2 is a process taking values in U (resp., V), progressively measurable with respect to the filtration ℱ, where ℱ=ℱr,r∈t1,t2 is the filtration generated by B and θ.
The set of all admissible controls for playerI (resp., player II) on time t,T is denoted by Ut,T (resp., Vt,T).Definition 2.
A nonanticipative strategy for playerI on t,T is a mapping α:V⟶U such that, for any ℱt-stopping time S and any v,v′∈Vt,T, if v≡v′ on t,S (with the notation t,S=s,ω∈0,T×Ω,t≤s≤Sω). In the same way, we define a nonanticipative strategy β:Ut,T⟶Vt,T for player II on t,T.
The set of all nonanticipative strategies for playerI (resp., player II) on t,T is denoted by At,T (resp., ℬt,T).
Now we give some properties about the lower and the upper value functionsW and U. The following lemma was established in [8], and the situation was slightly different. For the proof of this lemma, the readers can refer to [8].Lemma 6.
Under the assumptions (A3) and (A4), for allt,x,i∈0,T×Rn×ℳ, the value functions Wt,x,i and Ut,x,i are deterministic functions.From (4), (5), and (22), we get the properties of the lower value function W in the following.Lemma 7.
Under the assumptions (A3) and (A4), for allt,x,i∈0,T×Rn×ℳ and x,x′∈Rn, we have(i)
(23)Wt,x,i≤C1+x.(ii)
W⋅,x,i is 1/2-Hölder continuous with respect to t:(24)Wt,x,i−Wt′,x,i≤C1+xt−t′1/2.(iii)
(25)Wt,x,i−Wt,x′,i≤Cx−x′.The same properties hold true for the functionU.


## 3. Dynamic Programming Principles

The dynamic programming principle is one of the principal and most commonly used methods to solve the optimal control problem. In this section, we present the dynamic programming principle for a two-player zero-sum SDG with regime switching in the framework of BSDE on a finite time horizon. It will be used in the next section.We first introduce the backward stochastic semigroup. For given initial statet,x,i, a positive number δ≤T−t, for admissible control processes u⋅∈Ut,t+δ and v⋅∈Vt,t+δ, and a real value random variable η∈Ω,ℱt+δ,P;R, we define(26)Gt,t+δt,x,i;u,vη≔Y¯tt,x,i;u,v,where Y¯st,x,i;u,v,Z¯st,x,i;u,vs∈t,t+δ is the solution of the following BSDE with terminal time t+δ:(27)Y¯st,x,i;u,v=η+∫st+δfr,Xrt,x,i;u,v,Y¯rt,x,i;u,v,Z¯rt,x,i;u,v,θrt,i,ur,vrdr−∫st+δZ¯rt,x,i;u,vdBr,P−a.s.,s∈t,t+δ,and Xt,x,i;u,v is the solution of SDE (2). According to the uniqueness of the solution of the BSDE, we observe that for the solution Yt,x,i;u,v of BSDE (3), we have(28)Jt,x,i;u,v=EYtt,x,i;u,v|ℱt=EGt,Tt,x,i;u,vΦXTt,x,i;u,v,θTt,i|ℱt=EGt,t+δt,x,i;u,vYt+δt,x,i;u,v|ℱt=EGt,t+δt,x,i;u,vEYt+δt,x,i;u,v|ℱt+δ|ℱt=EGt,t+δt,x,i;u,vJt+δ,Xt+δt,x,i;u,v,θt+δt,i;u,v|ℱt.We now introduce the dynamic programming principle for the value functions of SDGs with regime switching.Proposition 1.
Under the assumptions (A3) and (A4), the following dynamic programming principle holds: for all0<δ≤T−t, x∈Rn, i∈ℳ,(29)Ut,x,i=esssupα∈At,t+δessinfv∈Vt,t+δEGt,t+δt,x,i;αv,vUt+δ,Xt+δt,x,i;αv,v,θt+δt,i|ℱt,Wt,x,i=essinfβ∈ℬt,t+δesssupu∈Ut,t+δEGt,t+δt,x,i;u,βuWt+δ,Xt+δt,x,i;u,βu,θt+δt,i|ℱt.Put(30)Wδt,x,i=essinfβ∈ℬt,t+δesssupu∈Ut,t+δEGt,t+δt,x,i;u,βuWt+δ,Xt+δt,x,i;u,βu,θt+δt,i|ℱt.We proceed with the proof thatWδt,x,i coincides with Wt,x,i into the following steps.Step 1. Let β∈ℬt,T be arbitrarily fixed. Then, given a u2∈Ut+δ,T, we define as follows the restriction β1 of β to Ut,t+δ:(31)β1u1≔βu1⊕u2|t,t+δ,u1∈Ut,t+δ,whereu1⊕u2≔u1χt,t+δ+u2χt+δ,T extends u1 to an element of Ut,T. Obviously, β1∈ℬt,t+δ. And, from the nonanticipative property of β we deduce that β1 is independent of the special choice of u2∈Ut+δ,T. Thus, from the definition of Wδt,x,i,(32)Wδt,x,i≤esssupu1∈Ut,t+δEGt,t+δt,x,i;u1,β1u1Wt+δ,Xt+δt,x,i;u1,β1u1,θt+δt,i|ℱt,and we use the notationℐδt,x,i;u,v≔EGt,t+δt,x,i;u,vWt+δ,Xt+δt,x,i;u,v,θt+δt,i|ℱt for some sequences ul1,l≥1⊂Ut,t+δ such that,(33)ℐδt,x,i;β1≔esssupu1∈Ut,t+δℐδt,x,i;u1,β1u1=supl≥1ℐδt,x,i;ul1,β1ul1,P−a.s.Letε>0 and set Γ˜l≔ℐδt,x,i;β1≤ℐδt,x,i;ul1,β1ul1+ε∈ℱt,l≥1. Construct Γ1≔Γ˜1,Γl≔Γ˜l,∪k=1l−1Γ˜k∈ℱt,l≥2. Certainly, Γll≥1 forms an Ω,ℱt-partition, and u1ε≔∑l≥1χΓlul1∈Ut,t+δ. Moreover, from the nonanticipativity of β1, we have β1u1ε=∑l≥1χΓlβ1ul1. According to the existence and uniqueness of the BSDEs, it follows that(34)ℐδt,x,i;u1ε,β1u1ε=∑l≥1χΓlℐδt,x,i;ul1,β1ul1,P−a.s.,forβ1∈ℬt,t+δ. Hence,(35)Wδt,x,i≤ℐδt,x,i;β1≤∑l≥1χΓlℐδt,x,i;ul1,β1ul1+ε=ℐδt,x,i;u1ε,β1u1ε+ε=EGt,t+δt,x,i;u1ε,β1u1εWt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i|ℱt+ε,P−a.s.We now focus on the intervalt+δ,T. Because β1⋅≔β⋅⊕u2∈ℬt,t+δ does not depend on u2∈Ut+δ,T, we can define β2u2≔βu1ε⊕u2|t+δ,T, for any u2∈Ut+δ,T. From β∈ℬt,T, we know that β2:Ut+δ,T⟶Vt+δ,T belongs to ℬt+δ,T. Thus, from the definition of Wt+δ,y,j, for any y,j∈Rn×ℳ,(36)Wt+δ,y,j≤esssupu2∈Ut+δ,TJt+δ,y,j;u2,β2u2,P−a.s.From Lemmas5 and 7, there exists a constant C∈R such that, for any u2∈Ut+δ,T, y,y′∈Rn,(37)iWt+δ,y,j−Wt+δ,y′,j≤Cy−y′,iiJt+δ,y,j;u2,β2u2−Jt+δ,y′,j;u2,β2u2≤Cy−y′,P−a.s.We can show by approximatingXt+δt,x,i;u1ε,β1u1ε that(38)Wt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i≤esssupu2∈Ut+δ,TJt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i;u2,β2u2,P−a.s.To estimate the right side of the latter inequality, we note that there exists some sequenceuk2,k≥1⊂Ut+δ,T such that(39)esssupu2∈Ut+δ,TJt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i;u2,β2u2=supk≥1Jt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i,uk2,β2uk2,P−a.s.Letε>0 and set △˜k≔esssupu2∈Ut+δ,TJt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i;u2,β2u2≤Jt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i;uk2,β2uk2+ε∈ℱt+δ,k≥1. Construct Δ1≔Δ˜1,Δk≔Δ˜k,∪j=1k−1Δ˜j∈ℱt+δ,k≥2. Certainly, Δkk≥1 forms an Ω,ℱt+δ-partition; moreover, u2ε≔∑k≥1χ△kuk2∈Ut+δ,T. Therefore, from the nonanticipativity of β2, we have β2u2ε=∑k≥1χ△kβ2uk2, and from the definition of β1,β2, we know that βu1ε⊕u2ε=β1u1ε⊕β2u2ε. According the existence and uniqueness of our BSDE, it follows that(40)Jt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i;u2ε,β2u2ε=EYt+δt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i;u2ε,β2u2ε|ℱt+δ=E∑k≥1χΔkYt+δt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i;ukε,β2ukε|ℱt+δ=∑k≥1χΔkJt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i;ukε,β2ukε,P−a.s.Therefore,(41)Wt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i≤esssupu2∈Ut+δ,TJt+δ,Xt+δt,x,i;u1ε,β1u1ε,θt+δt,i;u2,β2u2≤E∑k≥1χΔkYt+δt,x,i;u1ε⊕uk2,βu1ε⊕uk2|ℱt+δ+ε=EYt+δt,x,i;u1ε⊕u2ε,βu1ε⊕u2ε|ℱt+δ+ε=EYt+δt,x,i;uε,βuε|ℱt+δ+ε,P−a.s.,whereuε≔u1ε⊕u2ε∈Ut,T. From (35) and (41),(42)Wδt,x,i≤EGt,t+δt,x,i;u1ε,β1u1εEYt+δt,x,i;uε,βuε|ℱt+δ+ε|ℱt≤EGt,t+δt,x,i;u1ε,β1u1εEYt+δt,x,i;uε,βuε|ℱt+δ|ℱt+C+1ε=EGt,t+δt,x,i;uε,βuεEYt+δt,x,i;uε,βuε|ℱt+δ|ℱt+C+1ε=EGt,t+δt,x,i;uε,βuεYt+δt,x,i;uε,βuε|ℱt+C+1ε=EYtt,x,i;uε,βuε|ℱt+C+1ε≤esssupu∈Ut,TEYtt,x,i;u,βu|ℱt+C+1ε,P−a.s.Sinceβ∈ℬt,T has been arbitrarily chosen, we have (42) for all β∈ℬt,T. Thus,(43)Wδt,x,i≤essinfβ∈ℬt,Tesssupu∈Ut,TEYtt,x,i;u,βu|ℱt+C+1ε=Wt,x,i+C+1ε.Then, lettingε⟶0, we get Wδt,x,i≤Wt,x,i.Step 2. We now deal with the other case: Wt,x,i≤Wδt,x,i. From the definition of Wδt,x,i, we have(44)Wδt,x,i=essinfβ1∈ℬt,t+δesssupu1∈Ut,t+δEGt,t+δt,x,i;u1,β1u1Wt+δ,Xt+δt,x,i;u1,β1u1,θt+δt,i|ℱt=essinfβ1∈ℬt,t+δℐδt,x,i;β1,for some βl1,l≥1⊂ℬt,t+δ such that,(45)Wδt,x,i=infl≥1ℐδt,x,i;βl1,P−a.s.For anyε>0, we put Λ˜l≔ℐδt,x,i;βl1−ε≤Wδt,x,i∈ℱt,l≥1, Λ≔Λ˜1, and Λl≔Λ˜l,∪k=1l−1Λ˜k∈ℱt,l≥2. Certainly, Λll≥1 forms an Ω,ℱt-partition; moreover, β1ε≔∑l≥1χΛlβl1∈ℬt,t+δ. According the existence and uniqueness of our BSDE, we conclude that(46)ℐδt,x,i;u1,β1εu1=∑l≥1χΛlℐδt,x,i;u1,βl1u1,P−a.s,for all u1∈Ut,t+δ. Next,(47)Wδt,x,i≥∑l≥1χΛlℐδt,x,i;βl1−ε≥∑l≥1χΛlℐδt,x,i;u1,βl1u1−ε=ℐδt,x,i;u1,β1εu1−ε=EGt,t+δt,x,i;u1,β1εu1Wt+δ,Xt+δt,x,i;u1,β1εu1,θt+δt,i|ℱt−ε,P−a.s,for all u1∈Ut,t+δ.We now focus on the intervalt+δ,T. From the definition of Wt+δ,y,j, we deduce that, for any y,j∈Rn×ℳ, there exists βy,jε∈ℬt+δ,T such that(48)Wt+δ,y,j≥esssupu2∈Ut+δ,TJt+δ,y,j;u2,βy,jεu2−ε,P−a.s.Now consider a decomposition ofRn, namely, ∑l≥1Ol=Rn such that diamOl≤ε, for each l≥1. Take any yl,j∈Ol×ℳ,l≥1 fixed and define Xt+δt,x,i;u1,β1εu1¯≔∑l≥1,j∈ℳylχXt+δt,x,i;u1,β1εu1∈Ol,θt+δt,i=j. Clearly, we always have(49)Xt+δt,x,i;u1,β1εu1−Xt+δt,x,i;u1,β1εu1¯≤ε,everywhere on Ω, for each u1∈Ut,t+δ. Moreover, for every yl and j, there exists some βyl,jε∈ℬt+δ,T such that (48) holds, and clearly,(50)βu1ε≔∑l≥1,j∈ℳχXt+δt,x,i;u1,β1εu1∈Ol,θt+δt,i=jβyl,jε∈ℬt+δ,T.Now we can define the new strategyβεu≔β1εu1⊕βu1εu2,u∈Ut,T, where u1=u|t,t+δ,u2=u|t+δ,T. Obviously, βε:Ut,T⟶Vt,T.Next we shall show thatβε is nonanticipating: indeed, let S:Ω⟶t,T be an ℱ-stopping time and u,u′∈Ut,T be such that u≡u′ on t,S. Decomposing u,u′ into u1,u1′∈Ut,t+δ, u2,u2′∈Ut+δ,T such that u=u1⊕u2 and u′=u1′⊕u2′. We have u1≡u1′ on t,S∧t+δ since β1εu1≡β1εu1′ on t,S∧t+δ. On the other hand, u2≡u2′ on 〛t+δ,S∧t+δ〛⊂t+δ,T×S>t+δ, and on S>t+δ, we have Xt+δt,x,i;u1,β1εu1=Xt+δt,x,i;u1′,β1εu1′. Thus, from the definition, βu1ε=βu1′ε on S>t+δ and βu1εu2≡βu1′εu2′ on 〛t+δ,S∨t+δ〛. This yields βεu=β1εu1⊕βu1εu2≡β1εu1′⊕βu1′εu2′=βεu′ on t,S, from which it follows that βε∈ℬt,T.Fixu∈Ut,T arbitrarily and decompose into u1=u|t,t+δ∈Ut,t+δ, u2=u|t+δ,T∈Ut+δ,T. Then, from (37), (47), (i), and (49), we have(51)Wδt,x,i≥EGt,t+δt,x,i;u1,β1εu1Wt+δ,Xt+δt,x,i;u1,β1εu1,θt+δt,i|ℱt−ε≥EGt,t+δt,x,i;u1,β1εu1Wt+δ,Xt+δt,x,i;u1,β1εu1¯,θt+δt,i−Cε|ℱt−ε≥EGt,t+δt,x,i;u1,β1εu1Wt+δ,Xt+δt,x,i;u1,β1εu1¯,θt+δt,i|ℱt−Cε=EGt,t+δt,x,i;u1,β1εu1∑l≥1,j∈ℳχXt+δt,x,i;u1,β1εu1∈Ol,θt+δt,i=jWt+δ,yl,j|ℱt−Cε.From (37), (48), (ii), and (49), it follows that(52)Wδt,x,i≥EGt,t+δt,x,i;u1,β1εu1∑l≥1,j∈ℳχXt+δt,x,i;u1,β1εu1∈Ol,θt+δt,i=jJt+δ,yl,j;u2,βylεu2−ε|ℱt−Cε≥EGt,t+δt,x,i;u1,β1εu1∑l≥1,j∈ℳχXt+δt,x,i;u1,β1εu1∈Ol,θt+δt,i=jJt+δ,yl,j;u2,βylεu2|ℱt−Cε=EGt,t+δt,x,i;u1,β1εu1Jt+δ,Xt+δt,x,i;u1,β1εu1¯,θt+δt,i;u2,βu1εu2|ℱt−Cε≥EGt,t+δt,x,i;u1,β1εu1Jt+δ,Xt+δt,x,i;u1,β1εu1,θt+δt,i;u2,βu1εu2−Cε|ℱt−Cε≥EGt,t+δt,x,i;u1,β1εu1Jt+δ,Xt+δt,x,i;u1,β1εu1,θt+δt,i;u2,βu1εu2|ℱt−Cε=EGt,t+δt,x,i;u,βεuEYt+δt,x,i;u,βεu|ℱt+δ|ℱt−Cε=EGt,t+δt,x,i;u,βεuYt+δt,x,i;u,βεu|ℱt−Cε=EYtt,x,i;u,βεu|ℱt−Cε,P−a.s.,for any u∈Ut,T. Therefore, we obtain(53)Wδt,x,i≥esssupu∈Ut,TJt,x,i;u,βεu−Cε≥essinfβ∈ℬt,Tesssupu∈Ut,TJt,x,i;u,βu−Cε=Wt,x,i−Cε,P−a.s.Letε⟶0, and we have Wδt,x,i≥Wt,x,i.The proof is complete.


## 4. Viscosity Solution of Isaacs’ Equation: Existence and Uniqueness Theorem

In this section, based on the dynamic programming principle, we want to prove that the lower value functionWt,x,i introduced by (5) is the viscosity solution of (7), while the upper value function Ut,x,i defined by (6) is the viscosity solution of (8). Moreover, if Isaacs’ condition holds, i.e., H−=H+, then (7) and (8) coincide, and the uniqueness of viscosity solution implies that W=U, that is, the game admits a value.We first recall the definition of a viscosity solution of (7). The one for (8) can be defined in a similar wayDefinition 3.
A continuousW∈C0,T×Rnm is said to be a viscosity subsolution (resp., supersolution) of (7), if WT,x,i≤Φx,i (resp., WT,x,i≥Φx,i), for all x,i∈Rn×ℳ and if for all functions ϕ∈Cl,b30,T×Rn and t,x,i∈0,T×Rn×ℳ such that W⋅,⋅−ϕ⋅ attains its local maximum (resp., minimum) value zero at t,x, it has(54)∂ϕ∂tt,x+H−t,x,i,ϕ,Dϕ,D2ϕ+∑k≠iqikWt,x,k−Wt,x,i≤0,resp.,≥0,where W is called a viscosity solution if it is both a viscosity subsolution and a viscosity supersolution.Remark 1.
Cl,b30,T×Rn denotes the set of the real-valued functions that are continuously differentiable up to the third order and whose derivatives of order from 1 to 3 are bounded.
In the following, we prove that the lower value functionW is a viscosity solution of (7). We only focus on the lower value function W, and the results hold for the upper value function U in a similar procedure.Theorem 1.
Under the assumptions (A3) and (A4), the lower value functionWt,x,i is a viscosity solution of (7).First we prove some auxiliary lemmas. To abbreviate notations, for some arbitrarily chosen but fixedφ⋅,⋅∈Cl,b30,T×Rn,(55)Fs,x,y,z,i,u,v=∂φ∂ss,x+12TrσσTs,x,i,u,vD2φ+Dφ⋅bs,x,i,u,v+fs,x,y+φs,x,z+φs,x⋅σs,x,i,u,v,i,u,v+∑k≠iqikWs,x,k−Ws,x,i,where s,x,y,z,i,u,v∈0,T×Rn×R×Rd×ℳ×U×V and ∑k≠iqik<1, and we consider the following BSDE defined on the interval t,t+δ0<δ≤T−t:(56)−dYs1,u,v=Fs,Xst,x,i;u,v,Ys1,u,v,Zs1,u,v,i,us,vsds−Zs1,u,vdBs,Yt+δ1,u,v=0,where the process Xst,x,i;u,v has been introduced by (2), t,x,i∈0,T×Rn×ℳ is regarded as the initial state, Xs is the value of X at time s, and u∈Ut,t+δ,v∈Vt,t+δ.We can characterize the solution processY1,u,v as follows.Lemma 8.
For anys∈t,t+δ, t,x∈0,T×Rn, ϕ∈Cl,b30,T×Rn, we define φ∈C0,T×Rnm by φ⋅,⋅,i=ϕ⋅, and φ⋅,⋅,k=W⋅,⋅,k, and for k≠i, we have the following relationship:(57)EYs1,u,v|ℱt=EGs,t+δt,x,i;u,vφt+δ,Xt+δt,x,i;u,v,θt+δt,i−φs,Xst,x,i;u,v,θst,i|ℱt,P−a.s.Proof.
Gs,t+δt,x,i;u,vφt+δ,Xt+δt,x,i;u,v,θt+δt,i is defined with the help of the solution of the BSDE:(58)−dYsu,v=fs,Xst,x,i;u,v,Ysu,v,Zsu,v,i,us,vsds−Zsu,vdBs,Yt+δu,v=φt+δ,Xt+δt,x,i;u,v,θt+δt,i,s∈t,t+δ,by the following formula:(59)Gs,t+δt,x,i;u,vφt+δ,Xt+δt,x,i;u,v,θt+δt,i=Ysu,v.
Thus, we only need to proveYsu,v−φs,Xst,x,i;u,v,θst,i=Ys1,u,v. We have(60)EYsu,v−φs,Xst,x,i;u,v,θst,i|ℱt=EYs1,u,v|ℱt,by applying Dynkin’s formula to φs,Xst,x,i;u,v,θst,i. And for s=t+δ,(61)EYt+δu,v−φt+δ,Xt+δt,x,i;u,v,θt+δt,i|ℱt=EYt+δ1,u,v|ℱt=0.
Therefore, for anys∈t,t+δ, we get the desired result.We consider the following simple BSDE in whichXst,x,i;u,v is replaced by its deterministic initial value x:(62)−dYs2,u,v=Fs,x,Ys2,u,v,Zs2,u,v,i,us,vsds−Zs2,u,vdBs,Yt+δ2,u,v=0,s∈t,t+δ.Then, we have the following lemma.Lemma 9.
There is a constantC>0 independent of the control processes u, v and of δ>0, such that for every u∈Ut,t+δ, v∈Vt,t+δ,(63)EYt1,u,v−Yt2,u,v|ℱt≤Cδ3/2,P−a.s.Proof.
From Lemma4, we have the existence of some constant C>0 such that(64)Esups∈t,TXst,x,i;u,v2|ℱt≤C1+x2,combined with(65)Esups∈t,t+δXst,x,i;u,v−x2|ℱt≤2Esups∈t,t+δ∫tsbr,Xrt,x,i;u,v,i,ur,vrdr2|ℱt+2Esups∈t,t+δ∫tsσr,Xrt,x,i;u,v,i,ur,vrdBr2|ℱt,and we have(66)Esups∈t,t+δXst,x,i;u,v−x2|ℱt≤Cδ.
From (56) and (62), using Lemma 2, set(67)η1=η2=0,fs,y,z=Fs,Xst,x,i;u,v,y,z,i,us,vs,φ1s=0,φ2s=Fs,x,Ys2,u,v,Zs2,u,v,i,us,vs−Fs,Xst,x,i;u,v,Ys2,u,v,Zs2,u,v,i,us,vs.
It is easy to know thatg is Lipschitz with respect to y,z, and φ2s≤C1+x2Xst,x,i;u,v−x+Xst,x,i;u,v−x3 for s∈t,t+δ, t,x,i∈0,T×Rn×ℳ, u∈Ut,t+δ,v∈Vt,t+δ. Thus,(68)E∫tt+δYs1,u,v−Ys2,u,v2+Zs1,u,v−Zs2,u,v2ds|ℱt≤CE∫tt+δρ02Xst,x,i;u,v−xds|ℱt≤CδEsups∈t,t+δρ02Xst,x,i;u,v−x|ℱt≤Cδ2,where ρ0r=1+x2r+r3,r≥0. Therefore,(69)EYt1,u,v−Yt2,u,v|ℱt=EEYt1,u,v−Yt2,u,v|ℱt|ℱt=E|E∫tt+δFs,Xst,x,i;u,v,Ys1,u,v,Zs1,u,v,i,us,vs−Fs,x,Ys2,u,v,Zs2,u,v,i,us,vsds|ℱt||ℱt≤CE∫tt+δρ0Xst,x,i;u,v−x+Ys1,u,v−Ys2,u,v+Zs1,u,v−Zs2,u,vds|ℱt≤CE∫tt+δρ0Xst,x,i;u,v−xds|ℱt+Cδ1/2E∫tt+δYs1,u,v−Ys2,u,v2ds|ℱt1/2+Cδ1/2E∫tt+δZs1,u,v−Zs2,u,v2|ℱt1/2≤Cδ3/2.
The proof is complete.Lemma 10.
LetY0⋅ be the solution of the following ordinary differential equation:(70)−dY0s=F0s,x,Y0s,0,i,s∈t,t+δ,Y0t+δ=0,where(71)F0s,x,y,z,i=supu∈Uinfv∈VFs,x,y,z,i;u,v.Then,P−a.s.,(72)esssupu∈Ut,t+δessinfv∈Vt,t+δEYt2,u,v|ℱt=EY0t|ℱt.Proof.
We first introduce the function(73)F1s,x,y,z,i,u=infv∈VFs,x,y,z,i,u,v,where s,x,y,z,i,u∈0,T×Rn×R×Rd×ℳ×U. And we consider the following equation: for u∈Ut,t+δ,(74)−dYs3,u=F1s,x,Ys3,u,Zs3,u,i,usds−Zs3,udBs,Yt+δ3,u=0,s∈t,t+δ.
SinceF1s,x,y,z,i,us is Lipschitz in y,z, for every u∈Ut,t+δ, there exists a unique solution Y3,u,Z3,u to (74). Then, for every u∈Ut,t+δ,(75)EYt3,u|ℱt=essinfv∈Vt,t+δEYt2,u,v|ℱt,P−a.s.
In fact, from Lemma2 and the definition of F1, for every u∈Ut,t+δ, we have(76)EYt3,u|ℱt≤essinfv∈Vt,t+δEYt2,u,v|ℱt,P−a.s.
Moreover, there exists a measurable functionv4:t,T×Rn×R×Rd×ℳ×U⟶V such that(77)F1s,x,y,z,i,u=Fs,x,y,z,i,u,v4s,x,y,z,i,ufor anys,x,y,z,u.
Letv˜s4≔v4s,x,Ys3,u,Zs3,u,i,us,s∈t,t+δ, and we observe that v˜4∈Vt,t+δ, and(78)F1s,x,Ys3,u,Zs3,u,i,us=Fs,x,Ys3,u,Zs3,u,i,us,v˜s4,s∈t,t+δ.
Thus, from the uniqueness of the solution of the BSDE, it follows thatY3,u,Z3,u=Y2,u,v˜4,Z2,u,v˜4 and EYt3,u|ℱt=EYt2,u,v˜4|ℱt,P−a.s., for every u∈Ut,t+δ. Then, for all u∈Ut,t+δ,(79)EYt3,u|ℱt=essinfv∈Vt,t+δEYt2,u,v|ℱt,P−a.s.
Then, sinceF0s,x,y,z,i=supu∈Ut,t+δF1s,x,y,z,i,u, by a similar proof, we have(80)EY0t|ℱt=esssupu∈Ut,t+δEYt3,u|ℱt=esssupu∈Ut,t+δessinfv∈Vt,t+δEYt2,u,v|ℱt,P−a.s.
It uses the fact that (70) can be considered as a BSDE with the solution Ys,Zs=Y0s,0. So, the proof is complete.Lemma 11.
For allu∈Ut,t+δ, v∈Vt,t+δ, we have(81)E∫tt+δYs2,u,vds|ℱt+E∫tt+δZs2,u,vds|ℱt≤Cδ3/2,P−a.s.,where the constant C is independent of the control processes u, v and of δ>0.Proof.
SinceFs,x,⋅,⋅,i,u,v has a linear growth in y,z, uniformly in u,v, we get from Lemma 2 for some constant C independent of δ and the controls u, v,(82)Ys2,u,v2≤Cδ,E∫st+δZr2,u,vdr|ℱs≤Cδ,s∈t,t+δ.
Moreover, from equation (62),(83)Ys2,u,v≤E∫st+δFr,x,Yr2,u,v,Zr2,u,v,i,ur,vrdrℱs≤CE∫st+δ1+x2+Yr2,u,v+Zr2,u,vdr|ℱs≤Cδ+Cδ1/2E∫st+δZr2,u,v2dr|ℱs1/2≤Cδ,P−a.s.,s∈t,t+δ,and since(84)∫tt+δZs2,u,vdBs=∫tt+δFs,x,Ys2,u,v,Zs2,u,v,i,us,vsds−Yt2,u,v,we get(85)E∫tt+δZs2,u,v2ds|ℱt≤Cδ2.
Therefore,(86)E∫tt+δYs2,u,vds|ℱt+E∫tt+δZs2,u,vds|ℱt≤Cδ2+Cδ1/2E∫tt+δZs2,u,v2dr|ℱt1/2≤Cδ3/2,P−a.s.Proof of Theorem 1.
(i)
It is easy to know thatWT,x,i=Φx,i,x,i∈Rn×ℳ. We first will prove W is a viscosity supersolution. For fixed i∈ℳ, without loss of generality, we suppose that φt,x,i=Wt,x,i. According to Proposition 1,(87)φt,x,i=Wt,x,i=essinfβ∈ℬt,t+δesssupu∈Ut,t+δEGt,t+δt,x,i;u,βuWt+δ,Xt+δt,x,i;u,βu,θt+δt,i|ℱt,where0≤δ≤T−t and from W≥φ and the monotonicity property of Gt,t+δt,x,i;u,βu⋅, we have(88)essinfβ∈ℬt,t+δesssupu∈Ut,t+δEGt,t+δt,x,i;u,βuφt+δ,Xt+δt,x,i;u,βu,θt+δt,i−φt,x,i|ℱt≤0,P−a.s.From Lemma8,(89)essinfβ∈ℬt,t+δesssupu∈Ut,t+δEYt1,u,βu|ℱt≤0,P−a.s.Thus, from Lemma9, we obtain(90)essinfβ∈ℬt,t+δesssupu∈Ut,t+δEYt2,u,βu|ℱt≤Cδ3/2,P−a.s.Then, since(91)essinfv∈Vt,t+δEYt2,u,v|ℱt≤EYt2,u,βu|ℱt,β∈ℬt,t+δ,we have(92)esssupu∈Ut,t+δessinfv∈Vt,t+δEYt2,u,v|ℱt≤essinfβ∈ℬt,t+δesssupu∈Ut,t+δEYt2,u,βu|ℱt≤Cδ3/2,P−a.s,and Lemma10 implies Y0t≤Cδ3/2,P−a.s., where Y0 is the unique solution of (70). Thus,(93)Cδ1/2≥1δY0t=1δ∫tt+δF0s,x,Y0s,0,ids,δ>0,supu∈Uinfv∈VFt,x,0,0,i,u,v=F0t,x,0,0,i≤0,and from the definition ofF, we see that W is a viscosity supersolution of (7).(ii)
Now we proveW is a viscosity subsolution. For fixed i∈ℳ, without loss of generality, we suppose that φt,x,i=Wt,x,i. We must prove that(94)supu∈Uinfv∈VFt,x,0,0,i,u,v=F0t,x,0,0,i≥0.
We suppose that this is not true. Then, there exists someR>0 such that(95)F0t,x,0,0,i=supu∈Uinfv∈VFt,x,0,0,i,u,v≤−R<0,and we can find a measurable function ψ:U⟶V such that(96)Ft,x,0,0,i,u,ψu≤−34Rfor allu∈U.
And sinceF⋅,x,0,0,i,⋅,⋅ is uniformly continuous on 0,T×U×V, there exists some 0<Q≤T−t such that for all u∈U and s−t≤Q,(97)Fs,x,0,0,i,u,ψu≤−12R.
Moreover, due to Proposition1, for all 0≤δ≤Q,(98)φt,x,i=Wt,x,i=essinfβ∈ℬt,t+δessinfu∈Ut,t+δEGt,t+δt,x,i;u,βuWt+δ,Xt+δt,x,i;u,βu,θt+δt,i|ℱt,and similar to (i), from W≤φ and the monotonicity property of Gt,t+δt,x,i;u,βu⋅, we have(99)essinfβ∈ℬt,t+δesssupu∈Ut,t+δEGt,t+δt,x,i;u,βuφt+δ,Xt+δt,x,i;u,βu,θt+δt,i−φt,x,i|ℱt≥0,P−a.s.
Therefore, from Lemma8,(100)essinfβ∈ℬt,t+δesssupu∈Ut,t+δEYt1,u,βu|ℱt≥0,P−a.s.
Thus,(101)esssupu∈Ut,t+δEYt1,u,ψu|ℱt≥0,P−a.s.
Here, by settingψsuω=ψusω,s,ω∈t,T×Ω, we identify ψ as an element of ℬt,t+δ. Given any ε¿0, we can choose uε∈Ut,t+δ such that EYt1,uε,ψuε|ℱt≥−εδ. From Lemma 9, we get(102)EYt2,uε,ψuε|ℱt≥−Cδ3/2−εδ,P−a.s.
Moreover, from (62),(103)EYt2,uε,ψuε|ℱt=E∫tt+δFs,x,Ys2,uε,ψuε,Zs2,uε,ψuε,i,usε,ψusεds|ℱt.
We get from the Lipschitz property ofF in y,z, (99), and Lemma 11 that(104)EYt2,uε,ψuε|ℱt≤E∫tt+δCYs2,ue,ψuε+CZs2,uε,ψuε+Fs,x,0,0,i,usε,ψsuεds|ℱt≤Cδ3/2−12R,P−a.s.
Lettingδ↓0 and then ε↓0, we get that R≤0, which induces a contradiction. Then,(105)F0t,x,0,0,i=supu∈Uinfv∈VFt,x,0,0,i,u,v≥0,and from the definition of F, we see that W is a viscosity supersolution of (7). Therefore, from the above two steps, we derive that W is a viscosity solution of (7).Remark 2.
We can also prove that U is a viscosity solution of (8) similarly by the method of Theorem 1.We now study the uniqueness of the viscosity solution of Isaacs’ equation (7):(106)∂∂tht,x,i+H−t,x,i,h,Dh,D2h+∑k≠iqikWt,x,k−Wt,x,i=0,hT,x,i=Φx,i,where(107)H−t,x,i,p,A,u,v=supu∈Uinfv∈V12TrσσTt,x,i,u,v⋅A+bt,x,i,u,v,p+ft,x,i,r,σTt,x,i,u,vp,u,v,for t,x,i,r,p,A,u,v∈0,T×Rn×ℳ×R×Rn×Rn×n×U×V⟶R. The functions b,σ,f, and Φ are still supposed to satisfy (A3) and (A4), respectively.We will prove the uniqueness for (108) in the following space of continuous functions:(108)Θ=ϕ∈C0,T×Rn:∃A˜>0such thatlim|x|⟶∞ϕt,xexp−A˜logx2+11/22=0uniformly int∈0,T.This growth condition was introduced in [18]. And it was shown in [18] that this kind of growth condition is optimal for the uniqueness and can, in general, not be weakened. We adapt the idea developed in [18] to Isaacs’ equation (108) to prove the uniqueness of the viscosity solution in Θ. We only focus on equation (7) in Θ, and results hold for equation (8) in a similar procedure. For the proof of the uniqueness theorem, we need some auxiliary lemmas. Denoting by K a Lipschitz constant of ft,x,⋅,⋅,i,⋅,⋅, we have the following.Lemma 12.
LetW1∈Θ be a viscosity subsolution and W2∈Θ be a viscosity supersolution of (108). Then, the function h≔W1−W2 is a viscosity subsolution of the equation(109)∂∂tht,x+supu,v12TrσσTt,x,i,u,vD2h+Dh⋅bt,x,i,u,v+Kw+KDh⋅σt,x,i,u,v+∑k≠iqikht,x,k−ht,x,i=0,hT,x,i=0,t,x,i∈0,T×Rn×ℳ.The proof of this lemma follows directly from Lemma 3.7 in [18]; it is only slightly different.Lemma 13.
LetW1∈Θ be a viscosity subsolution and W2∈Θ be a viscosity supersolution of (108). Then, the function h˜≔maxi∈ℳW1t,x,i−W2t,x,i is a viscosity subsolution of the equation(110)∂∂th˜t,x+supu,v,i12TrσσTt,x,i,u,vD2h˜+Dh˜⋅bt,x,i,u,v+Kh˜+KDh˜⋅σt,x,i,u,v=0,h˜T,x=0,t,x∈0,T×Rn.Proof.
Fort0,x0∈0,T×Rn, without loss of generality, we suppose that h˜t0,x0=W1t0,x0,i−W2t0,x0,i, i.e.,(111)ht0,x0,i≥ht0,x0,k,∀k≠i,where h≔W1−W2. Hence,(112)∑k≠iqikht0,x0,k−ht0,x0,i≤0.
Obviously,h˜T,x≤0, for all x∈Rn. Moreover, from (114), for all functions ϕ∈Cl,b30,T×Rn,(113)∂∂tϕt0,x0+supu,v,i12TrσσTt0,x0,i,u,vD2ϕ+Dϕ⋅bt0,x0,i,u,v+Kϕ+KDϕ⋅σt0,x0,i,u,v≥∂∂tϕt0,x0+supu,v12TrσσTt0,x0,i,u,vD2ϕ+Dϕ⋅bt0,x0,i,u,v+Kϕ+KDϕ⋅σt0,x0,i,u,v≥∂∂tϕt0,x0+supu,v12TrσσTt0,x0,i,u,vD2ϕ+Dϕ⋅bt0,x0,i,u,v+Kϕ+KDϕ⋅σt0,x0,i,u,v+∑k≠iqikϕt0,x0,k−ϕt0,x0,i.
On the other hand, for anyϕ∈Cl,b30,T×Rn such that h˜−ϕ attains its local maximum value zero at t0,x0, we have that h⋅,⋅,i−ϕ⋅,i attains its local maximum value zero at t0,x0. Since h is a viscosity subsolution of (111), we have(114)∂∂tϕt0,x0+supu,v12TrσσTt0,x0,i,u,vD2ϕ+Dϕ⋅bt0,x0,i,u,v+Kϕ+KDϕ⋅σt0,x0,i,u,v+∑k≠iqikϕt0,x0,k−ϕt0,x0,i≥0.
Consequently,(115)∂∂tϕt0,x0+supu,v,i12TrσσTt0,x0,i,u,vD2ϕ+Dϕ⋅bt0,x0,i,u,v+Kϕ+KDϕ⋅σt0,x0,i,u,v≥0.
It means thath˜ is a viscosity subsolution of equation (112).Similar to the proof of Theorem 6.1 in [8], we can prove that h˜≤0, i.e., W1t,x,i≤W2t,x,i. Then, we have the uniqueness theorem.Theorem 2.
Under the assumptions (A3) and (A4), the viscosity solution of (108) is unique in Θ.Remark 3.
Since the lower value functionW is of at most linear growth, it belongs to Θ, and it is easy to know that W is the unique viscosity solution in Θ of (108). And we can prove that the upper value function U is the unique viscosity solution in Θ of (8).Remark 4.
If Isaacs’ condition holds, that is, if for anyt,x,i,r,p∈0,T×Rn×ℳ×R×Rn, H+t,x,i,r,p,A,u,v=H−t,x,i,r,p,A,u,v, then (8) and (108) coincide, i.e., (7) and (8) coincide, and from the uniqueness in Θ of viscosity solution, it follows that W=U which means the associated SDG has a value.



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*Source: 9930142-2021-06-10.xml*