# The Dynamic Programming Method of Stochastic Differential Game for Functional Forward-Backward Stochastic System

Authors: Shaolin Ji; Chuanfeng Sun; Qingmeng Wei
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is devoted to a stochastic differential game (SDG) of decoupled functional forward-backward stochastic differential equation (FBSDE). For our SDG, the associated upper and lower value functions of the SDG are defined through the solution of controlled functional backward stochastic differential equations (BSDEs). Applying the Girsanov transformation method introduced by Buckdahn and Li (2008), the upper and the lower value functions are shown to be deterministic. We also generalize the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations to the path-dependent ones. By establishing the dynamic programming principal (DPP), we derive that the upper and the lower value functions are the viscosity solutions of the corresponding upper and the lower path-dependent HJBI equations, respectively.


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## Body

## 1. Introduction

The theory of backward stochastic differential equations (BSDEs) has been studied widely since Pardoux and Peng [1] first introduced the nonlinear BSDEs in 1990. BSDEs have got applications in many fields, such as, stochastic control (see Peng [2]), stochastic differential games (SDGs) (see Hamadene and Lepeltier [3], Hamadene et al. [4]), mathematical finance (see El Karoui et al. [5]) and partial differential equation (PDE) theory (see Peng [6, 7]), and so forth.In the aspect of finance, the BSDE theory presents a simple formulation of stochastic differential utilities introduced by Duffie and Epstein [8]. When the generator g of a BSDE does not depend on z, the solution Y is just the recursive utility presented in [8]. From the view of BSDE, by studying some important properties (such as, comparison theorem) of BSDEs, El Karoui et al. [5] gave the more general class of recursive utilities and their properties. And later the recursive optimal control problems whose cost functionals are described by the solution of BSDE are studied widely. Peng [7] obtained the Bellman’s dynamic programming principle (DPP) for this kind of problem and proved the value function to be a viscosity solution of one kind of quasi-linear second order PDE, that is, Hamilton-Jacobi-Bellman (HJB) equation. Later, for the recursive optimal control problem introduced by a BSDE under Markovian framework, by introducing the notion of backward semigroup of BSDE, in Peng [9], the Bellman’s DPP is derived and the value function is proved to be a viscosity solution of a generalized HJB equation.By now, the DPP with related HJB equation has become a powerful approach to solving optimal control and game problems (see [10–13]). In [10], Buckdahn and Li studied a recursive SDG problem and interpreted the relationship between the controlled system and the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation. A point is worthy to mention: in order to derive the DPP, they introduced a Girsanov transformation method to prove the value functions are deterministic which is different from the method developed in Peng [9].There really exist some systems which are modeled only by stochastic systems whose evolutions depend on the past history of the states. Based on this phenomenon, Ji and Yang [14] investigated a controlled system governed by a functional forward-backward stochastic differential equation (FBSDE) and proved the value function to be the viscosity solution of the related path-dependent HJB equation.In this paper, inspired by [10, 14], we will investigate the SDG problems of the functional FBSDEs. Precisely, the dynamics of our SDGs is described by the following functional SDE:
(1)dXγt;u,v(s)=b(Xsγt;u,v,u(s),v(s))ds+σ(Xsγt;u,v,u(s),v(s))dB(s),s∈[t,T],Xtγt,u,v=γt.
And the cost functional J(γt;u,v) is defined as Yγt;u,v(t) which is the solution of the following functional BSDE:
(2)dYγt;u,v(s)=-f(Xsγt;u,v,Yγt;u,v(s),Zγt;u,v(s),u(s),v(s)Xsγt;u,v)ds+Zγt;u,v(s)dB(s),Yγt,u,v(T)=Φ(XTγt;u,v),s∈[t,T],
where γt is a path on [0,t]. The driver f and Φ can be interpreted as the running cost and the terminal cost, respectively. Also, they depend on the past history of the dynamics. Equations (1) and (2) compose a decoupled functional FBSDE. The concrete conditions on b,σ,f,Φ are shown in the later section.In the context, we adopt the strategy against control form. The cost functionalJ(γt;u,v) is explained as a payoff for player I and as a cost for player II. The aim of this paper is to show the following lower and upper value functions:
(3)W(γt)≔essinfβ∈ℬt,Tesssupu∈𝒰t,TJ(γt;u,β(u)),U(γt)≔esssupα∈𝒜t,Tessinfv∈𝒱t,TJ(γt;α(v),v)
are the viscosity solutions of the following path-dependent HJBI equations, respectively,
(4)DtW(γt)+supu∈Uinfv∈VH(γt,W,DxW,DxxW,u,v)=0,W(γT)=Φ(γT),γT∈Λ,DtU(γt)+infv∈Vsupu∈UH(γt,U,DxU,DxxU,u,v)=0,U(γt)=Φ(γT),γT∈Λ,
where
(5)H(γt,y,p,X,u,v)=12tr(σσT(γt,u,v)X)+p·b(γt,u,v)+f(γt,y,p·σ(γt,u,v),u,v),
where (γt,y,p,X)∈Λ×ℝ×ℝd×𝕊d(𝕊d denotes the set of d×d symmetric matrices).To solve the above SDG, we need the functional Itô’s calculus and path-dependent PDEs which are recently introduced by Dupire [15] (for a recent account of this theory, the reader may consult [16–18]. And under the framework of functional Itô’s calculus, for the non-Markovian BSDEs, Peng and Wang [19] derived a nonlinear Feynman-Kac formula for classical solutions of path-dependent PDEs. For the further development, the readers may refer to [20, 21]).In this paper, we apply the Girsanov transformation method in Buckdahn and Li [10] to prove the determinacy of the value functions, which is different from the method introduced by Peng [7, 9]. Making use of this method and the functional Itô’s calculus (introduced by Dupire [15] and developed by Cont and Fournié [16–18]), we complete the study of the zero-sum two-player SDGs in the non-Markovian case and present the lower and upper value functions of our SDG are the viscosity solutions of the corresponding path-dependent HJBI equations, respectively.Different from the HJBI equations developed for stochastic delay systems, we establish the DPP and derive the HJBI equation in the new framework of functional Itô’s calculus.This paper is organized as follows. Section2 recalls the functional Itô’s calculus and the well-known results of BSDEs we will use later. In Section 3, we formulate our SDGs and get the corresponding DPP. Based on the obtained DPP, in Section 4 we derive the main result of the paper: the lower and upper value functions are the viscosity solutions of the associated path-dependent HJBI equations, respectively. And we add the proof for the DPP in the appendix.


## 2. Preliminaries

### 2.1. Functional Itô’s Calculus

We present some preliminaries for functional Itô’s calculus introduced firstly by Dupire [15]. Here we follow the notations in [15].LetT>0 be fixed. For each t∈[0,T], we denote Λt the set of càdlàg functions from [0,t] to ℝd.Forγ∈ΛT, denote γ(s) by the value of γ at time s∈[0,T]. Thus γ=(γ(s))0≤s≤T is a càdlàg process on [0,T] and its value at time s is γ(s). γt=(γ(s))0≤s≤t∈Λt is the path of γ up to time t. We denote Λ=⋃t∈[0,T]Λt. For each γt∈Λ and x∈ℝd, γt(s) is denoted by the value of γt at s∈[0,t] and γtx:=(γt(s)0≤s<t,γt(t)+x) which is also an element in Λt.We now introduce a distance onΛ. Let 〈·,·〉 and |·| denote the inner product and norm in ℝd. For each 0≤t,t-≤T and γt,γt-∈Λ, we set
(6)∥γt∥≔sups∈[0,t]|γt(s)|,∥γt-γ-t-∥≔sups∈[0,t∨t-]|γt(s∧t)-γ-t-(s∧t-)|,d∞(γt,γ-t-)≔sups∈[0,t∨t-]|γt(s∧t)-γ-t-(s∧t-)|+|t-t-|.
It is obvious that Λt is a Banach space with respect to ∥·∥. Since Λ is not a linear space, d∞ is not a norm.Definition 1.
A functionalu:Λ↦ℝ is Λ-continuous at γt∈Λ, if for any ε>0 there exists δ>0 such that for each γ-t-∈Λ with d∞(γt,γ-t-)<δ, one has |u(γt)-u(γ-t-)|<ε.u is said to be Λ-continuous if it is Λ-continuous at each γt∈Λ.Definition 2.
Letv:Λ↦ℝ and γt∈Λ be given. If there exists p∈ℝd, such that
(7)v(γtx)=v(γt)+〈p,x〉+o(|x|),asx→0,x∈ℝd.
Then we say that v is (vertically) differentiable at t and denote the gradient of Dxv(γt)=p·v is said to be vertically differentiable in Λ, if Dxv(γt) exists for each γt∈Λ. The Hessian Dxxv(γt) can be defined similarly. It is an S(d)-valued function defined on Λ, where S(d) is the space of all d×d symmetric matrices.For eachγt∈Λ, we denote γt,s(r):=γt(r)1[0,t)(r)+γt(t)1[t,s](r),r∈[0,s]. It is clear that γt,s∈Λs.Definition 3.
For a givenγt∈Λ, if one has
(8)v(γt,s)=v(γt)+a(s-t)+o(|s-t|),ass→t,s≥t,
then we say that v(γt) is (horizontally) differentiable in t at γt and denote Dtv(γt)=a. v is said to be horizontally differentiable in Λ if Dtv(γt) exists for each γt∈Λ.Definition 4.
Defineℂj,k(Λ) as the set of function v:=(v(γt))γt∈Λ defined on Λ which are j times horizontally and k times vertically differentiable in Λ such that all these derivatives are Λ-continuous.The following is about the functional Itô’s formula which was firstly obtained by Dupire [15] and then developed by Cont and Fournié [18] for a more general formulation.Theorem 5 (functional Itô’s formula).
Let(Ω,ℱ,  (ℱt)t∈[0,T],P) be a probability space, if X is a continuous semimartingale and v is in ℂ1,2(Λ), then for any t∈[0,T),
(9)v(Xt)-v(X0)=∫0t‍Dsv(Xs)ds+∫0t‍Dxv(Xs)dX(s)+12∫0t‍Dxxv(Xs)d〈X〉(s),P-a.s.


### 2.2. BSDEs

In this section, we recollect some important results which will be used in our SDG problems.Let(Ω,ℱ,P) be the Wiener space, where Ω is the set of continuous functions from [0,T] to ℝd starting from 0 (Ω=C0([0,T];ℝd)), T is an arbitrarily fixed real time horizon, ℱ is the completed Borel σ-algebra over Ω, and P is the Wiener measure. Let B be the canonical process: B(ω,s)=ωs,s∈[0,T],ω∈Ω. We denote by 𝔽={ℱs,0≤s≤T} the natural filtration generated by {B(t)}t≥0 and augmented by all P-null sets, that is, ℱs=σ{B(r),r≤s}∨𝒩P,s∈[0,T];ℱts=σ(B(r)-B(t),t≤r≤s)∨𝒩P, where 𝒩P is the set of all P-null subsets. First we present two spaces of processes as follows:
(10)𝒮2(0,T;ℝn)≔{[sup0≤t≤T|ψ(t)|2](ψ(t))0≤t≤Tℝn-valued𝒮2(0,T;ℝn)≔𝔽-adaptedcontinuousprocess:E[sup0≤t≤T|ψ(t)|2]<+∞};ℋ2(0,T;ℝn)≔{∫0T‍(ψ(t))0≤t≤Tℝn-valuedℋ2(0,T;ℝn)≔𝔽-progressivelymeasurableprocess:ℋ2(0,T;ℝn)≔∥ψ∥2=E[∫0T‍|ψ(t)|2dt]<+∞}.Considerg:Ω×[0,T]×ℝ×ℝd→ℝ, for every (y,z) in ℝ×ℝd, (g(t,y,z))t∈[0,T] is progressively measurable and satisfies the following conditions:(A1)
there exists a constantC≥0 such that, P-a.s., for all t∈[0,T],y1,y2∈ℝ,z1,z2∈ℝd,
(11)|g(t,y1,z1)-g(t,y2,z2)|≤C(|y1-y2|+|z1-z2|),(A2)
g(·,0,0)∈ℋ2(0,T;ℝ).In the following, we suppose the driverg of a BSDE satisfies (A1) and (A2).Lemma 6.
Under the assumptions(A1) and (A2), for any random variable ξ∈L2(Ω,ℱT,P;ℝ), the BSDE
(12)y(t)=ξ+∫tTg(s,y(s),z(s))ds-∫tTz(s)dB(s),0≤t≤T
has a unique adapted solution
(13)(y(t),z(t))t∈[0,T]∈𝒮2(0,T;ℝ)×ℋ2(0,T;ℝd).The readers may refer to Pardoux and Peng [1] for the above well-known existence and uniqueness results on BSDEs.Lemma 7 (comparison theorem).
Given two coefficientsg1, g2 satisfying (A1), (A2), and two terminal values ξ1,ξ2∈L2(Ω,ℱT,P;ℝ), by (y1,z1) and (y2,z2) one denotes the solution of a BSDE with the data (ξ1,g1) and (ξ2,g2), respectively. Then one has the following.(i)
Ifξ1≥ξ2 and g1≥g2, P-a.s., then y1(t)≥y2(t), P-a.s., for all t∈[0,T].(ii)
(Strict monotonicity) If, in addition to (i), one also assumes thatP(ξ1>ξ2)>0, then P(y1(t)>y2(t))>0, 0≤t≤T, and, in particular, y1(0)>y2(0).With the notations in Lemma7, we assume that, for some g:Ω×[0,T]×ℝ×ℝd→ℝ satisfying (A1) and (A2), the drivers gi have the following form:
(14)gi(s,yi(s),zi(s))=g(s,yi(s),zi(s))+φi(s),dsdP-a.e.,i=1,2,
where φi∈ℋ2(0,T;ℝ). Then, the following results hold true for all terminal values ξ1,ξ2∈L2(Ω,ℱT,P;ℝ).Lemma 8.
The difference of the solutions(y1,z1) and (y2,z2) of BSDE (12) with the data (ξ1,g1) and (ξ2,g2), respectively, satisfies the following estimate:
(15)|y1(t)-y2(t)|2+12E[∫tTeβ(s-t)(|y1(s)-y2(s)|2‍+|z1(s)-z2(s)|2)ds∣ℱt∫tT‍]≤E[eβ(T-t)|ξ1-ξ2|2∣ℱt]+E[∫tT‍eβ(s-t)|φ1(s)-φ2(s)|2ds∣ℱt],P-a.s.,∀0≤t≤T,whereβ=16(1+C2).Proof.
The reader may refer to Proposition 2.1 in El Karoui et al. [5] or Theorem 2.3 in Peng [9] for the details.


## 2.1. Functional Itô’s Calculus

We present some preliminaries for functional Itô’s calculus introduced firstly by Dupire [15]. Here we follow the notations in [15].LetT>0 be fixed. For each t∈[0,T], we denote Λt the set of càdlàg functions from [0,t] to ℝd.Forγ∈ΛT, denote γ(s) by the value of γ at time s∈[0,T]. Thus γ=(γ(s))0≤s≤T is a càdlàg process on [0,T] and its value at time s is γ(s). γt=(γ(s))0≤s≤t∈Λt is the path of γ up to time t. We denote Λ=⋃t∈[0,T]Λt. For each γt∈Λ and x∈ℝd, γt(s) is denoted by the value of γt at s∈[0,t] and γtx:=(γt(s)0≤s<t,γt(t)+x) which is also an element in Λt.We now introduce a distance onΛ. Let 〈·,·〉 and |·| denote the inner product and norm in ℝd. For each 0≤t,t-≤T and γt,γt-∈Λ, we set
(6)∥γt∥≔sups∈[0,t]|γt(s)|,∥γt-γ-t-∥≔sups∈[0,t∨t-]|γt(s∧t)-γ-t-(s∧t-)|,d∞(γt,γ-t-)≔sups∈[0,t∨t-]|γt(s∧t)-γ-t-(s∧t-)|+|t-t-|.
It is obvious that Λt is a Banach space with respect to ∥·∥. Since Λ is not a linear space, d∞ is not a norm.Definition 1.
A functionalu:Λ↦ℝ is Λ-continuous at γt∈Λ, if for any ε>0 there exists δ>0 such that for each γ-t-∈Λ with d∞(γt,γ-t-)<δ, one has |u(γt)-u(γ-t-)|<ε.u is said to be Λ-continuous if it is Λ-continuous at each γt∈Λ.Definition 2.
Letv:Λ↦ℝ and γt∈Λ be given. If there exists p∈ℝd, such that
(7)v(γtx)=v(γt)+〈p,x〉+o(|x|),asx→0,x∈ℝd.
Then we say that v is (vertically) differentiable at t and denote the gradient of Dxv(γt)=p·v is said to be vertically differentiable in Λ, if Dxv(γt) exists for each γt∈Λ. The Hessian Dxxv(γt) can be defined similarly. It is an S(d)-valued function defined on Λ, where S(d) is the space of all d×d symmetric matrices.For eachγt∈Λ, we denote γt,s(r):=γt(r)1[0,t)(r)+γt(t)1[t,s](r),r∈[0,s]. It is clear that γt,s∈Λs.Definition 3.
For a givenγt∈Λ, if one has
(8)v(γt,s)=v(γt)+a(s-t)+o(|s-t|),ass→t,s≥t,
then we say that v(γt) is (horizontally) differentiable in t at γt and denote Dtv(γt)=a. v is said to be horizontally differentiable in Λ if Dtv(γt) exists for each γt∈Λ.Definition 4.
Defineℂj,k(Λ) as the set of function v:=(v(γt))γt∈Λ defined on Λ which are j times horizontally and k times vertically differentiable in Λ such that all these derivatives are Λ-continuous.The following is about the functional Itô’s formula which was firstly obtained by Dupire [15] and then developed by Cont and Fournié [18] for a more general formulation.Theorem 5 (functional Itô’s formula).
Let(Ω,ℱ,  (ℱt)t∈[0,T],P) be a probability space, if X is a continuous semimartingale and v is in ℂ1,2(Λ), then for any t∈[0,T),
(9)v(Xt)-v(X0)=∫0t‍Dsv(Xs)ds+∫0t‍Dxv(Xs)dX(s)+12∫0t‍Dxxv(Xs)d〈X〉(s),P-a.s.


## 2.2. BSDEs

In this section, we recollect some important results which will be used in our SDG problems.Let(Ω,ℱ,P) be the Wiener space, where Ω is the set of continuous functions from [0,T] to ℝd starting from 0 (Ω=C0([0,T];ℝd)), T is an arbitrarily fixed real time horizon, ℱ is the completed Borel σ-algebra over Ω, and P is the Wiener measure. Let B be the canonical process: B(ω,s)=ωs,s∈[0,T],ω∈Ω. We denote by 𝔽={ℱs,0≤s≤T} the natural filtration generated by {B(t)}t≥0 and augmented by all P-null sets, that is, ℱs=σ{B(r),r≤s}∨𝒩P,s∈[0,T];ℱts=σ(B(r)-B(t),t≤r≤s)∨𝒩P, where 𝒩P is the set of all P-null subsets. First we present two spaces of processes as follows:
(10)𝒮2(0,T;ℝn)≔{[sup0≤t≤T|ψ(t)|2](ψ(t))0≤t≤Tℝn-valued𝒮2(0,T;ℝn)≔𝔽-adaptedcontinuousprocess:E[sup0≤t≤T|ψ(t)|2]<+∞};ℋ2(0,T;ℝn)≔{∫0T‍(ψ(t))0≤t≤Tℝn-valuedℋ2(0,T;ℝn)≔𝔽-progressivelymeasurableprocess:ℋ2(0,T;ℝn)≔∥ψ∥2=E[∫0T‍|ψ(t)|2dt]<+∞}.Considerg:Ω×[0,T]×ℝ×ℝd→ℝ, for every (y,z) in ℝ×ℝd, (g(t,y,z))t∈[0,T] is progressively measurable and satisfies the following conditions:(A1)
there exists a constantC≥0 such that, P-a.s., for all t∈[0,T],y1,y2∈ℝ,z1,z2∈ℝd,
(11)|g(t,y1,z1)-g(t,y2,z2)|≤C(|y1-y2|+|z1-z2|),(A2)
g(·,0,0)∈ℋ2(0,T;ℝ).In the following, we suppose the driverg of a BSDE satisfies (A1) and (A2).Lemma 6.
Under the assumptions(A1) and (A2), for any random variable ξ∈L2(Ω,ℱT,P;ℝ), the BSDE
(12)y(t)=ξ+∫tTg(s,y(s),z(s))ds-∫tTz(s)dB(s),0≤t≤T
has a unique adapted solution
(13)(y(t),z(t))t∈[0,T]∈𝒮2(0,T;ℝ)×ℋ2(0,T;ℝd).The readers may refer to Pardoux and Peng [1] for the above well-known existence and uniqueness results on BSDEs.Lemma 7 (comparison theorem).
Given two coefficientsg1, g2 satisfying (A1), (A2), and two terminal values ξ1,ξ2∈L2(Ω,ℱT,P;ℝ), by (y1,z1) and (y2,z2) one denotes the solution of a BSDE with the data (ξ1,g1) and (ξ2,g2), respectively. Then one has the following.(i)
Ifξ1≥ξ2 and g1≥g2, P-a.s., then y1(t)≥y2(t), P-a.s., for all t∈[0,T].(ii)
(Strict monotonicity) If, in addition to (i), one also assumes thatP(ξ1>ξ2)>0, then P(y1(t)>y2(t))>0, 0≤t≤T, and, in particular, y1(0)>y2(0).With the notations in Lemma7, we assume that, for some g:Ω×[0,T]×ℝ×ℝd→ℝ satisfying (A1) and (A2), the drivers gi have the following form:
(14)gi(s,yi(s),zi(s))=g(s,yi(s),zi(s))+φi(s),dsdP-a.e.,i=1,2,
where φi∈ℋ2(0,T;ℝ). Then, the following results hold true for all terminal values ξ1,ξ2∈L2(Ω,ℱT,P;ℝ).Lemma 8.
The difference of the solutions(y1,z1) and (y2,z2) of BSDE (12) with the data (ξ1,g1) and (ξ2,g2), respectively, satisfies the following estimate:
(15)|y1(t)-y2(t)|2+12E[∫tTeβ(s-t)(|y1(s)-y2(s)|2‍+|z1(s)-z2(s)|2)ds∣ℱt∫tT‍]≤E[eβ(T-t)|ξ1-ξ2|2∣ℱt]+E[∫tT‍eβ(s-t)|φ1(s)-φ2(s)|2ds∣ℱt],P-a.s.,∀0≤t≤T,whereβ=16(1+C2).Proof.
The reader may refer to Proposition 2.1 in El Karoui et al. [5] or Theorem 2.3 in Peng [9] for the details.


## 3. A DPP for Stochastic Differential Games of Functional FBSDEs

In this section, we consider the SDGs of functional FBSDEs.First we introduce the background of SDGs. Suppose that the control state spacesU,V are compact metric spaces. 𝒰 (resp., 𝒱) is the control set of all U (resp., V)-valued 𝔽-progressively measurable processes for the first (resp., second) player. If u∈𝒰 (resp., v∈𝒱), we call u (resp., v) an admissible control.Let us give the following mappings:(16)b:Λ×U×V→ℝn,σ:Λ×U×V→ℝn×d,f:Λ×ℝ×ℝd×U×V→ℝ.For given admissible controlsu(·)∈𝒰, v(·)∈𝒱, and t∈[0,T],γt∈Λ, we consider the following functional forward-backward stochastic system:
(17)dXγt;u,v(s)=b(Xsγt;u,v,u(s),v(s))ds+σ(Xsγt;u,v,u(s),v(s))dB(s),s∈[t,T],dYγt;u,v(s)=-f(Xsγt;u,v,Yγt;u,v(s),Zγt;u,v(s),u(s),v(s))ds+Zγt;u,v(s)dB(s),Xtγt,u,v=γt,Yγt,u,v(T)=Φ(XTγt;u,v).(H) (i)
For allt∈[0,T], u∈U,  v∈V, xt∈Λ,  y∈ℝ,  z∈ℝd, b(xt,u,v),σ(xt,u,v), and f(xt, y,z,u,v) are ℱt-measurable.(ii)
There exists a constantC>0, such that, for all t∈[0,T], u∈U,  v∈V, for any xt1,xt2∈Λ,
(18)|b(xt1,u,v)-b(xt2,u,v)|+|σ(xt1,u,v)-σ(xt2,u,v)|≤C∥xt1-xt2∥,(19)|b(xt,u,v)|+|σ(xt,u,v)|≤C(1+∥xt∥),foranyxt∈Λ.(iii)
There exists a constantC>0, such that for all t∈[0,T], u∈U,v∈V, for any xt1,xt2∈Λ,  y1,y2∈ℝ,  z1,z2∈ℝd(20)|f(xt1,y1,z1,u,v)-f(xt2,y2,z2,u,v)|≤C(∥xt1-xt2∥+|y1-y2|+|z1-z2|),|Φ(xT1)-Φ(xT2)|≤C∥xT1-xT2∥,|f(xt,0,0,u,v)|≤C(1+∥xt∥),|Φ(xT)|≤C(1+∥xT∥),for anyxt∈Λ.Theorem 9.
Under the assumption(H), there exists a unique solution (X,Y,Z)∈𝒮2(0,T;ℝn)×𝒮2(0,T;ℝ)×ℋ2(0,T;ℝd) solving (17).We recall the subspaces of admissible controls and the definitions of admissible strategies as follows, which are similar to [10].Definition 10.
An admissible control processu=(ur)r∈[t,s] (resp., v=(vr)r∈[t,s]) for Player I (resp., II) on [t,s] is an ℱr-progressively measurable, U (resp., V)-valued process. The set of all admissible controls for Player I (resp., II) on [t,s] is denoted by 𝒰t,s (resp., 𝒱t,s). If P{u≡u-,a.e.,in[t,s]}=1, one will identify both processes u and u- in 𝒰t,s. Similarly one interprets v≡v- on [t,s] in 𝒱t,s.Definition 11.
A nonanticipative strategy for Player I on[t,s](t<s≤T) is a mapping α:𝒱t,s→𝒰t,s such that, for any 𝔽-stopping time S:Ω→[t,s] and any v1,v2∈𝒱t,s, with v1≡v2 on [[t,S]], it holds that α(v1)≡α(v2) on [[t,S]]. Nonanticipative strategies for Player II on [t,s], β:𝒰t,s→𝒱t,s, are defined similarly. The set of all nonanticipative strategies α:𝒱t,s→𝒰t,s for Player I on [t,s] is denoted by 𝒜t,s. The set of all nonanticipative strategies β:𝒰t,s→𝒱t,s for Player II on [t,s] is denoted by ℬt,s. (Recall that [[t,S]]={(r,ω)∈[0,T]×Ω,t≤r≤S(ω).)For given processesu(·)∈𝒰t,T,v(·)∈𝒱t,T, initial data t∈[0,T],γt∈Λ, the cost functional is defined as follows:
(21)J(γt;u,v)≔Yγt;u,v(t),γt∈Λ,
where the process Yγt;u,v is defined by functional FBSDE (17).Forγt∈Λ, the lower and the upper value functions of our SDGs are defined as
(22)W(γt)≔essinfβ∈ℬt,Tesssupu∈𝒰t,TJ(γt;u,β(u)),(23)U(γt)≔esssupα∈𝒜t,Tessinfv∈𝒱t,TJ(γt;α(v),v).As we know, the essential infimum and essential supremum on a family of random variables are still random variables. But by applying the method introduced by Buckdahn and Li [10], we get W(γt) and U(γt) are deterministic.Proposition 12.
For anyt∈[0,T],γt∈Λ,W(γt) is a deterministic function in the sense that W(γt)=E[W(γt)],P-a.s.Proof.
LetH denote the Cameron-Martin space of all absolutely continuous elements h∈Ω whose derivative h˙ belongs to L2([0,T];ℝd).
For anyh∈H, we define the mapping τhω≔ω+h,ω∈Ω. It is easy to check that τh:Ω→Ω is a bijection, and its law is given by P∘[τh]-1=exp{∫0T‍h˙(s)dB(s)-(1/2)∫0T‍|h˙(s)|2ds}P. For any fixed t∈[0,T], set Ht={h∈H∣h(·)=h(·∧t)}. The proof can be separated into the following four steps.
(
1
) For all u∈𝒰t,T,v∈𝒱t,T,h∈Ht,J(γt;u,v)(τh)=J(γt;u(τh),v(τh)),P-a.s.
First, we make the transformation for the functional SDE as follows:(24)Xγt;u,v(s)∘(τh)=(γt(t)+∫ts‍b(Xrγt;u,v,u(r),v(r))dr+∫ts‍σ(Xrγt;u,v,u(r),v(r))dB(r))∘(τh)=γt(t)+∫ts‍b(Xrγt;u,v(τh),u(τh)(r),v(τh)(r)Xrγt;u,v)dr+∫ts‍σ(Xrγt;u,v(τh),u(τh)(r),v(τh)(r)Xrγt;u,v)dB(r),Xγt;u(τh),v(τh)(s)=γt(t)+∫ts‍b(Xrγt;u(τh),v(τh),u(τh)(r),v(τh)(r)Xrγt;u(τh),v(τh))dr+∫ts‍σ(Xrγt;u(τh),v(τh),u(τh)(r),v(τh)(r)Xrγt;u(τh),v(τh))dB(r),
then, from the uniqueness of the solution of the functional SDE, we get
(25)Xγt;u,v(s)(τh)=Xγt;u(τh),v(τh)(s),foranys∈[t,T],P-a.s.
Similarly, using the transformation to the BSDE in (17) and comparing the obtained equation with the BSDE obtained from (17) by replacing the transformed control process u(τh),v(τh) for u,v, due to the uniqueness of the solution of functional BSDE, we obtain
(26)Yγt;u,v(s)(τh)=Yγt;u(τh),v(τh)(s),for  anys∈[t,T],P-a.s.,Zγt;u,v(s)(τh)=Zγt;u(τh),v(τh)(s),dsdP-a.e.on[0,T]×Ω.
Hence
(27)J(γt;u,v)(τh)=J(γt;u(τh),v(τh)),P-a.s.
(
2
) For β∈ℬt,T,h∈Ht, let βh(u):=β(u(τ-h))(τh),u∈𝒰t,T. Then, βh∈ℬt,T.
Obviously,βh:𝒰t,T→𝒱t,T. And it is nonanticipating. In fact, given an 𝔽-stopping time S:Ω→[t,T] and u1,u2∈𝒰t,T, with u1≡u2 on [[t,S]]. Accordingly, u1(τ-h)≡u2(τ-h) on [[t,S(τ-h)]]. Thus,
(28)βh(u1)=β(u1(τ-h))(τh)=β(u2(τ-h))(τh)=βh(u2)on[[t,S]].
(
3
) For any h∈Ht, and β∈ℬt,T, we have
(29){esssupu∈𝒰t,TJ(γt;u,β(u))}(τh)=esssupu∈𝒰t,T{J(γt;u,β(u))(τh)},P-a.s.
In fact, for convenience, setting I(γt;β)≔esssupu∈𝒰t,TJ(γt;u,β(u)),β∈ℬt,T, we know I(γt;β)≥J(γt;u,β(u)). Then I(γt;β)(τh)≥J(γt;u,β(u))(τh),P-a.s.,for all u∈𝒰t,T. Therefore,
(30){esssupu∈𝒰t,TJ(γt;u,β(u))}(τh)≥esssupu∈𝒰t,T{J(γt;u,β(u))(τh)},P-a.s.
From the definition of essential supremum, for any random variableξ which satisfies ξ≥J(γt;u,β(u))(τh), we have ξ(τ-h)≥J(γt;u,β(u)),P-a.s. for all u∈𝒰t,T. So ξ(τ-h)≥I(γt;β),P-a.s., that is, ξ≥I(γt;β)(τh),P-a.s. Thus,
(31)J(γt;u,β(u))(τh)≥{esssupu∈𝒰t,TJ(γt;u,β(u))}(τh),P-a.s.,for  anyu∈𝒰t,T.
Therefore,
(32)esssupu∈𝒰t,T{J(γt;u,β(u))(τh)}≥{esssupu∈𝒰t,TJ(γt;u,β(u))}(τh),P-a.s.
From above we get
(33){esssupu∈𝒰t,TJ(γt;u,β(u))}(τh)=esssupu∈𝒰t,T{J(γt;u,β(u))(τh)},P-a.s.
(
4
) Under the Girsanov transformation τh, W(γt) is invariant, that is,
(34)W(γt)(τh)=W(γt),P-a.s.,foranyh∈H.
In fact, we can prove {essinfβ∈ℬt,TI(γt;β)}(τh)=essinfβ∈ℬt,T{I(γt;β)(τh)},P-a.s., for all h∈Ht, which is similar to the above step. From the above three steps, for all h∈Ht, we get
(35)W(γt)(τh)={essinfβ∈ℬt,Tesssupu∈𝒰t,TJ(γt;u,β(u))}(τh),=essinfβ∈ℬt,Tesssupu∈𝒰t,T{J(γt;u,β(u))(τh)},=essinfβ∈ℬt,Tesssupu∈𝒰t,T{J(γt;u(τh),βh(u(τh)))},=essinfβ∈ℬt,Tesssupu∈𝒰t,TJ(γt;u,β(u)),=W(γt),P-a.s..
Note that {u(τh)∣u(·)∈𝒰t,T}=𝒰t,T and {βh∣β∈ℬt,T}=ℬt,T have been used in the above equalities. So, for any h∈Ht,W(γt)(τh)=W(γt),P-a.s. Thanks to W(γt) is ℱt-measurable, this relation holds true for all h∈H.
To finish the proof, we also need the auxiliary lemma as follows.Lemma 13.
Letζ be a random variable defined over the classical Wiener space (Ω,ℱT,P), such that ζ(τh)=ζ,P-a.s., for any h∈H. Then ζ=Eζ,P-a.s.Proof.
From Lemma13 in Buckdahn and Li [10], we know for any A∈ℬ(ℝ),φ∈L2([0,T];ℝd),
(36)E[1{ζ∈A}exp{∫0T‍φ(s)dB(s)}]=E[1{ζ∈A}]E[exp{∫0T‍φ(s)dB(s)}].
For any φ=∑i=1Nφi1(ti-1,ti], where φi∈ℝd, for 0≤i≤N and {ti}i=0N is a finite partition of [0,T], from (36),
(37)E[1{ζ∈A}exp(∑i=1N‍φiB(ti)-B(ti-1))]=E[1{ζ∈A}]E[exp∑i=1N‍φi(B(ti)-B(ti-1))].
Therefore, for any nonnegative integer ki(38)E[1{ζ∈A}∏i=1N‍(B(ti)-B(ti-1))ki]=E[1{ζ∈A}]E[∏i=1N(B(ti)-B(ti-1))ki].
So, for any polynomial function Q, we have
(39)E[1{ζ∈A}Q(B(t1)-B(t0),…,B(tn)-B(tn-1))]=E[1{ζ∈A}]E[Q(B(t1)-B(t0),…,B(tn)-B(tn-1))].
Furthermore, for any Q∈Cb(ℝn), we still have (39). Combining the arbitrariness of A∈ℬ(ℝ), we obtain ζ is independent of (B(t1)-B(t0),…,B(tn)-B(tn-1)), for all partition of [0,T]. Therefore, ζ is independent of ℱT which implies ζ is independent of itself, that is Eζ=ζ,P-a.s.In Ji and Yang [14], they proved the following estimates.Lemma 14.
Under the assumption(H), there exists some constant C>0 such that, for any t∈[0,T],γt,γ-t∈Λ,u(·)∈𝒰,v(·)∈𝒱,
(40)E[sups∈[t,T]|Xγt,u,v(s)|2∣ℱt]≤C(1+∥γt∥2),E[sups∈[t,T]|Xγt,u,v(s)-Xγ-t,u,v(s)|2∣ℱt]≤C∥γt-γ-t∥2,E[sups∈[t,T]|Yγt,u,v(s)|2+∫tT|Zγt,u,v(s)|2ds∣ℱt]≤C(1+∥γt∥2),E[sups∈[t,T]|Yγt,u,v(s)-Yγ-t,u,v(s)|2+∫tT|Zγt,u,v(s)-Zγ-t,u,v(s)|2ds∣ℱt]≤C∥γt-γ-t∥2.From the definition ofW(γt) and Lemma 14, we have the following property.Lemma 15.
There exists some constantC>0 such that, for all 0≤t≤T,γt,γ-t∈Λ,
(41)(i)|W(γt)-W(γ-t)|≤C∥γt-γ-t∥,(ii)|W(γt)|≤C(1+∥γt∥).Now we adopt Peng’s notion of stochastic backward semigroup (which was first introduced by Peng [9] to prove the DPP for stochastic control problems) to discuss a generalized DPP for our SDG (17), (22). First we define the family of backward semigroups associated with FBSDE (17).For givent∈[0,T],γt∈Λ, a number δ∈(0,T-t], admissible control processes u(·)∈𝒰t,t+δ,v(·)∈𝒱t,t+δ, we set
(42)Gs,t+δγt;u,v[η]:=Y~γt;u,v(s),s∈[t,t+δ],
where η∈L2(Ω,ℱtt+δ,P;ℝ), (Y~γt;u,v(s),Z~γt;u,v(s))t≤s≤t+δ solves the following functional FBSDE on [t,t+δ]:
(43)dY~γt;u,v(s)=-f(s,Xsγt;u,v,Y~γt;u,v(s),Z~γt;u,v(s),u(s),v(s))ds+Z~γt;u,v(s)dB(s),Y~γt;u,v(t+δ)=η,s∈[t,t+δ].
Also, we have
(44)Gt,Tγt;u,v[Φ(XTγt;u,v)]=Gt,t+δγt;u,v[Yγt;u,v(t+δ)].Theorem 16.
Suppose(H) holds true, the lower value function W(γt) satisfies the following DPP: for any t∈[0,T],γt∈Λ,δ>0,
(45)W(γt)=essinfβ∈ℬt,t+δesssupu∈𝒰t,t+δGt,t+δγt;u,β(u)[W(Xt+δγt;u,β(u))].The proof is given in the appendix.


## 4. Viscosity Solutions of Path-Dependent HJBI Equation

Now we study the following path-dependent PDEs:(46)DtW(γt)+H-(γt,W,DxW,DxxW)=0,W(γT)=Φ(γT),γT∈Λ,(47)DtU(γt)+H+(γt,U,DxU,DxxU)=0,U(γt)=Φ(γT),γT∈Λ,
where
(48)H-(γt,W,DxW,DxxW)=supu∈Uinfv∈VH(γt,W,DxW,DxxW,u,v),H+(γt,U,DxU,DxxU)=infv∈Vsupu∈UH(γt,U,DxU,DxxU,u,v),H(γt,y,p,X,u,v)=12tr(σσT(γt,u,v)X)+p·b(γt,u,v)+f(γt,y,p.σ(γt,u,v),u,v),
where (γt,y,p,X)∈Λ×ℝ×ℝd×𝕊d (𝕊d denotes the set of d×d symmetric matrices).We will show that the value functionW(γt) (resp., U(γt)) defined in (22) (resp., (23)) is a viscosity solution of the corresponding equation (46) (resp., (47)). First we give the definition of viscosity solution for this kind of PDEs. For more information on viscosity solution, the reader is referred to  Crandall et al. [22].Definition 17.
A real-valuedΛ-continuous function W∈ℂ(Λ) is called(i)
a viscosity subsolution of (46) if for any δ>0,  Γ∈ℂl,b1,2(Λ),γt∈Λ satisfying Γ≥W on ⋃0≤s≤δΛt+s and Γ(γt)=W(γt), one has
(49)DtΓ(γt)+H-(γt,Γ,DxΓ,DxxΓ)≥0,(ii)
a viscosity supersolution of (46) if for any δ>0,Γ∈ℂl,b1,2(Λ),γt∈Λ satisfying Γ≤W on ⋃0≤s≤δΛt+s and Γ(γt)=W(γt), one has
(50)DtΓ(γt)+H-(γt,Γ,DxΓ,DxxΓ)≤0,(iii)
a viscosity solution of (46) if it is both a viscosity sub- and supersolution of (46).Theorem 18.
Assume(H) holds, the lower value function W is a viscosity solution of path-dependent HJBI Equation (46) on Λ, the upper value function U is a viscosity solution of (47).First we prove some helpful lemmas. For some fixedΓ∈ℂl,b1,2(Λ), denote
(51)F(γs,y,z,u,v)=DsΓ(γs)+12tr(σσT(γs,u,v)DxxΓ(γs))+DxΓ(γs)·b(γs,u,v)+f(γs,y+Γ(γs),z+DxΓ(γs)·σ(γs,u,v),u,v),
where (γs,y,z,u,v)∈Λ×ℝ×ℝd×U×V.Consider the following BSDE:(52)-dY1,u,v(s)=F(Xsγt;u,v,Y1,u,v(s),Z1,u,v(s),u(s),v(s))ds-Z1,u,v(s)dB(s),Y1,u,v(t+δ)=0,s∈[t,t+δ].Lemma 19.
For everys∈[t,t+δ], one has the following:
(53)Y1,u,v(s)=Gs,t+δγt;u,v[Γ(Xt+δγt;u,v)]-Γ(Xsγt;u,v),P-a.s.Proof.
NoteGs,t+δγt;u,v[Γ(Xt+δγt;u,v)] is defined as Gs,t+δγt;u,v[Γ(Xt+δγt;u,v)]=Yu,v(s) through the following BSDE:
(54)-dYu,v(s)=f(Xsγt;u,v,Yu,v(s),Zu,v(s),u(s),v(s))ds-dYu,v(s)=-Zu,v(s)dB(s),Yu,v(t+δ)=Γ(Xt+δγt;u,v),s∈[t,t+δ].
Using Itô’s formula to Γ(Xsγt;u,v), we have
(55)d(Yu,v(s)-Γ(Xsγt;u,v))=dY1,u,v(s).
Combined with Yu,v(t+δ)-Γ(Xt+δγt;u,v)=0=Y1,u,v(t+δ), we get the desired result.Now consider the following BSDE:(56)-dY2,u,v(s)=F(γt,Y2,u,v(s),Z2,u,v(s),u(s),v(s))ds-dY2,u,v(s)=-Z2,u,v(s)dB(s),(57)Y2,u,v(t+δ)=0,s∈[t,t+δ].Then, we have the following lemma.Lemma 20.
For everyu∈𝒰t,t+δ,v∈𝒱t,t+δ, one has
(58)|Y1,u,v(t)-Y2,u,v(t)|≤Cδ3/2,P-a.s.
where C is independent of the control processes u,v.Proof.
From Lemma14, we know there exists some constant C>0 such that
(59)E[sups∈[t,T]|Xγt;u,v(s)|2∣ℱt]≤C(1+∥γt∥2),
combined with
(60)E[sups∈[t,t+δ]|Xγt;u,v(s)-γt(t)|2∣ℱt]≤2E[sups∈[t,t+δ]|∫tsb(Xrγt;u,v,u(r),v(r))dr|2∣ℱt]+2E[sups∈[t,t+δ]|∫tsσ(Xrγt;u,v,u(r),v(r))dB(r)|2∣ℱt]
we have
(61)E[sups∈[t,t+δ]|Xγt;u,v(s)-γt(t)|2∣ℱt]≤Cδ.
From (52) and (54), using Lemma 8, set
(62)ξ1=ξ2=0,g(s,y,z)=F(Xsγt;u,v,y,z,us,vs),φ1(s)=0,φ2(s)=F(γt,Y2,u,v(s),Z2,u,v(s),us,vs)-F(Xsγt;u,v,Y2,u,v(s),Z2,u,v(s),us,vs).
Denote ρ0(r)=(1+|x|2)r, due to b,σ,f are Lipishtiz and that they are of linear growth, Γ∈ℂb1,2, |φ2(s)|≤ρ0(|Xsγt;u,v-γt(t)|), we have
(63)E[∫tt+δ‍(|Y1,u,v(s)-Y2,u,v(s)|2+|Z1,u,v(s)-Z2,u,v(s)|2)ds∣ℱt∫tt+δ‍]≤E[∫tt+δ‍ρ02(|Xsγt;u,v-γt(t)|)ds∣ℱt]≤CδE[sups∈[t,t+δ]ρ02(|Xsγt;u,v-γt(t)|)∣ℱt]≤Cδ2.
So,
(64)|Y1,u,v(t)-Y2,u,v(t)|=|E[(Y1,u,v(s)-Y2,u,v(s))∣ℱt]|=|E[∫tt+δ(F(Xsγt;u,v,Y1,u,v(s),Z1,u,v(s),u(s),v(s))-F(γt,Y2,u,v(s),Z2,u,v(s),u(s),v(s)))ds∣ℱt∫tt+δ]|≤CE[∫tt+δ‍[ρ0(|Xsγt;u,v-γt|)+|Y1,u,v(s)-Y2,u,v(s)|+|Z1,u,v(s)-Z2,u,v(s)||Xsγt;u,v-γt|]ds∣ℱt∫tt+δ‍]≤CE[∫tt+δ‍ρ0(|Xsγt;u,v-γt|)ds∣ℱt]+Cδ1/2E[∫tt+δ‍|Y1,u,v(s)-Y2,u,v(s)|2ds∣ℱt]1/2+Cδ1/2E[∫tt+δ‍|Z1,u,v(s)-Z2,u,v(s)|2ds∣ℱt]1/2≤Cδ3/2.Lemma 21.
DenoteY0(·) by the solution of the following ordinary differential equation:
(65)-dY0(s)=F0(γt,Y0(s),0)ds,s∈[t,t+δ],Y0(t+δ)=0,
where F0(γt,y,z)≔supu∈Uinfv∈VF(γt,y,z,u,v). Then, P-a.s.,
(66)Y0(t)=esssupu∈𝒰t,t+δessinfv∈𝒱t,t+δY2,u,v(t).Proof.
First we define a function as follows:(67)F1(γt,y,z,u)≔infv∈VF(γt,y,z,u,v),(γt,y,z,u)∈Λ×ℝ×ℝd×U.
Consider the following equation:
(68)-dY3,u(s)=F1(γt,Y3,u(s),Z3,u(s),u(s))ds-Z3,u(s)dB(s),s∈[t,t+δ],Y3,u(t+δ)=0,
according to Lemma 6, for every u∈𝒰t,t+δ, there exists a unique (Y3,u,Z3,u) solving (68). Also,
(69)Y3,u(t)=essinfv∈𝒱t,t+δY2,u,v(t),P-a.s.foreveryu∈𝒰t,t+δ.
In fact, according to the definition of F1 and Lemma 7, we have
(70)Y3,u(t)≤Y2,u,v(t),P-a.s.foranyv∈𝒱t,t+δ,foreveryu∈𝒰t,t+δ.
On the other side, we have the existence of a measurable functionv4:ℝ×ℝd×U→V such that
(71)F1(γt,y,z,u)=F(γt,y,z,u,v4(y,z,u)),foranyy,z,u.
Set v~4(s)=v4(Y3,u(s),Z3,u(s),u(s)),s∈[t,t+δ], we know v~4∈𝒱t,t+δ. Therefore, (Y3,u,Z3,u)=(Y2,u,v~4,Z2,u,v~4) which is due to the uniqueness of the solution of (68). In particular, Y3,u(t)=Y2,u,v~4(t),P-a.s. for every u∈𝒰t,t+δ. Hence, Y3,u(t)=essinfv∈𝒱t,t+δY2,u,v(t),P-a.s. for every u∈𝒰t,t+δ.
Similarly, fromF0(γt,y,z)=supu∈UF1(γt,y,z,u), we also derive
(72)Y0(t)=esssupu∈𝒰t,t+δY3,u(t)=esssupu∈𝒰t,t+δessinfv∈𝒱t,t+δY2,u,v(t),P-a.s.Lemma 22.
For everyu∈𝒰t,t+δ,v∈𝒱t,t+δ, one has
(73)E[∫tt+δ(|Y2,u,v(s)|+|Z2,u,v(s)|)ds∣ℱt]≤Cδ3/2,P-a.s.,
where the constant C is independent of the control processes u,v.Proof.
Due toF(γt,·,·,u,v) is linear growth in (y,z), uniformly in (u,v), from Lemma 8, we know there exists a constant C>0 which does not depend on δ nor on the controls u and v, such that, P-a.s.(74)|Y2,u,v(s)|2≤Cδ,E[∫st+δ|Z2,u,v(r)|2dr∣ℱs]≤Cδ.
Moreover, from (54), we have
(75)|Y2,u,v(s)|≤E[∫st+δF(γt,Y2,u,v(r),Z2,u,v(r),u(r),v(r))dr∣ℱs]≤CE[∫st+δ(1+∥γt∥2+|Y2,u,v(r)|+|Z2,u,v(r)|)dr∣ℱs]≤Cδ+Cδ1/2E[∫st+δ|Z2,u,v(r)|2dr∣ℱs]≤Cδ,s∈[t,t+δ],
and applying Itô’s formula, we get E[∫tt+δ‍|Z2,u,v(s)|2ds∣ℱt]≤Cδ2,P-a.s. Therefore,
(76)E[∫tt+δ(|Y2,u,v(s)|+|Z2,u,v(s)|)ds∣ℱt]≤Cδ2+Cδ1/2(E[∫tt+δ|Z2,u,v(s)|2dr∣ℱt])1/2≤Cδ3/2,P-a.s.Now we continue the proof of Theorem18.Proof.
(
1
) First we will prove W(γt) is a viscosity supersolution. Given γt∈Λ, for any δ>0, suppose Γ(γt)=W(γt) and W≥Γ on ⋃0≤s≤δΛt+s.
According to the DPP (Theorem16), we have
(77)Γ(γt)=W(γt)=essinfβ∈ℬt,t+δesssupu∈𝒰t,t+δGt,t+δγt;u,β(u)[W(Xt+δγt;u,β(u))].
From W≥Γ on ⋃0≤s≤δΛt+s as well as the monotonicity property of Gt,t+δγt;u,β(u)[·] (Lemma 7) we have
(78)essinfβ∈ℬt,t+δesssupu∈𝒰t,t+δGt,t+δγt;u,β(u)[Γ(Xt+δγt;u,β(u))]-Γ(γt)≤0.
By Lemma 19, we have
(79)essinfβ∈ℬt,t+δesssupu∈𝒰t,t+δY1,u,β(u)(t)≤0,P-a.s.
Thus, from Lemma 20,
(80)essinfβ∈ℬt,t+δesssupu∈𝒰t,t+δY2,u,β(u)(t)≤Cδ3/2,P-a.s.
Therefore, from essinfv∈𝒱t,t+δY2,u,v(t)≤Y2,u,β(u)(t),β∈ℬt,t+δ, we have
(81)esssupu∈𝒰t,t+δessinfv∈𝒱t,t+δY2,u,v(t)≤essinfβ∈ℬt,t+δesssupu∈𝒰t,t+δY2,u,β(u)(t)≤Cδ3/2,P-a.s.
Thus, by Lemma 21, Y0(t)≤Cδ3/2,P-a.s., where Y0(·) solves the ODE (65). Consequently,
(82)Cδ1/2≥1δY0(t)=1δ∫tt+δ‍F0(γt,Y0(s),0)ds,δ>0,(83)supu∈Uinfv∈VF(γt,0,0,u,v)=F0(γt,0,0)≤0.
Obviously, according to the definition of F, W is a viscosity supersolution of (46).
(2) Now we proveW is a viscosity subsolution.
Givenγt∈Λ, for any δ>0, suppose Γ(γt)=W(γt) and Γ≥W on ⋃0≤s≤δΛt+s.
We need to prove(84)supu∈𝒰infv∈VF(γt,0,0,u,v)=F0(γt,0,0)≥0.
Suppose it does not hold true. So we have some θ>0 such that
(85)F0(γt,0,0)=supu∈𝒰infv∈VF(γt,0,0,u,v)≤-θ<0,
and there exists a measurable function ψ:U→V such that
(86)F(γt,0,0,u,ψ(u))≤-34θ,∀u∈U.
Moreover, from the DPP (Theorem16),
(87)Γ(γt)=W(γt)=essinfβ∈ℬt,t+δesssupu∈𝒰t,t+δGt,t+δγt;u,β(u)[W(Xt+δγt,u,β(u))].
Again from W≤Γ on ⋃0≤s≤δΛt+s as well as the monotonicity property of Gt,t+δγt;u,β(u)[·] (Lemma 7) we get
(88)essinfβ∈ℬt,t+δesssupu∈𝒰t,t+δGt,t+δγt;u,β(u)[Γ(Xt+δu,β(u))]-Γ(γt)≥0,P-a.s.Then, similar to (1), from the definition of backward semigroup, we have
(89)essinfβ∈ℬt,t+δesssupu∈𝒰t,t+δY1,u,β(u)(t)≥0,P-a.s.,
in particular,
(90)esssupu∈𝒰t,t+δY1,u,ψ(u)(t)≥0,P-a.s.
Setting ψs(u)(ω)=ψ(u(s)(ω)),(s,ω)∈[t,T]×Ω,ψ can be regarded as an element of ℬt,t+δ. Given any ε>0 we can select uε∈𝒰t,t+δ satisfying Y1,uε,ψ(uε)(t)≥-εδ.
From Lemma20 we have
(91)Y2,uε,ψ(uε)(t)≥-Cδ3/2-εδ,P-a.s.
Moreover, from (54) we have
(92)Y2,uε,ψ(uε)(t)=E[∫tt+δF(γt,Y2,uε,ψ(uε)(s),Z2,uε,ψ(uε)(s),‍uε(s),ψ(uε(s)))ds∣ℱt∫tt+δ‍]≤E[∫tt+δ‍((γt,0,0,uε(s),ψ(uε(s)))C|Y2,uε,ψ(uε)(s)|+C|Z2,uε,ψ(uε)(s)|+F(γt,0,0,uε(s),ψ(uε(s))))ds∣ℱt∫tt+δ‍]≤Cδ3/2-34θδ,P-a.s.
From (91) and (92), we have -Cδ1/2-ε≤Cδ1/2-(3/4)θ,P-a.s. Letting δ↓0, and then ε↓0, we get θ≤0, which produces a contradiction. Consequently,
(93)supu∈Uinfv∈VF(γt,0,0,u,v)=F0(γt,0,0)≤0.
According to the definition of F, W is a viscosity supersolution of (46). At last, from the above two steps, we derive that W is a viscosity solution of (46).
The similar argument forU, we also get that U is a viscosity solution of (47).



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*Source: 958920-2013-03-06.xml*