Stochastic differential game of functional forward-backward stochastic system and related path-dependent HJBI equation

0.25 MB·English

by  Shaolin Ji

Stochastic differential game of functional forward-backward stochastic
system and related path-dependent HJBI equation
2
1 Shaolin Ji∗ Qingmeng Wei†
0
2
Shandong University Shandong University

p
e
S
7 Abstract: This paper is devoted to a stochastic differential game of functional forward-backward
2
stochastic differential equation (FBSDE, for short). The associated upper and lower value functions

] of the stochastic differential game are defined by controlled functional backward stochastic differential
C
equations (BSDEs, for short). Applying the Girsanov transformation method introduced by Buckdahn
O
andLi[1], the upper andthe lowervalue functions areshownto be deterministic. We alsogeneralizethe
.
h
Hamilton-Jacobi-Bellman-Isaacs(HJBI,forshort)equationstothepath-dependentones. Byestablishing
t
a
m the dynamicprogrammingprincipal(DPP, forshort),the upper andthe lowervaluefunctions areshown
to be the viscosity solutions of the corresponding upper and the lower path-dependent HJBI equations,
[

respectively.
1
v Keywords: Stochastic differential game; Functional forward-backward stochastic differential equa-
9
tion; Path-dependent Hamilton-Jacobi-Bellman-Isaacs equation; Dynamic programming principle; Vis-
6
1 cosity solution.
6
.
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0 1 Introduction
2
1
: The theory of backward stochastic differential equations (BSDEs, for short) has been studied widely
v
i sincePardouxandPeng [14]firstintroducedthe nonlinearBSDEs in1990. BSDEs havegotapplications
X
in many fields, such as, stochastic control (see Peng [17]), stochastic differential games (see Hamadene,
r
a
Lepeltier [10], Hamadene, Lepeltier and Peng [11]), mathematical finance (see El Karoui, Peng and
Quenez [12]) and partial differential equation theory (see Peng [18, 19]), etc.
In the aspect of finance, the BSDE theory presents a simple formulation of stochastic differential
utilities introduced by Duffie and Epstein [6]. When the generator g of a BSDE does not depend on z,
the solution Y is just the recursive utility presented in [6]. From the view of BSDE, by studying some
important properties (such as, comparison theorem) of BSDEs, El Karoui, Peng and Quenez [12] 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 [19]
obtained the Bellman’s dynamic programming principle (DPP) for this kind of problem and proved the
∗Institute for Financial Studies and Institute of Mathematics, Shandong University, Jinan 250100, China. Email:
[email protected],Fax: [phone]
†InstituteofMathematics, ShandongUniversity,Jinan250100, China. Email: [email protected]
1
value function to be a viscosity solution of one kind of quasi-linear second order PDE, i.e., Hamilton-
Jacobi-Bellman(HJB) equation. Later,for the recursiveoptimal controlproblemintroducedby a BSDE
under Markovian framework, by introducing the notion of backward semigroup of BSDE, in Peng [20],
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 [1] [9], [23], [24]). In [1], Buckdahn and Li studied a recursive stochas-
tic differential game problem and interpreted the relationship between the controlled system and the
Hamilton-Jacobi-Bellman-Isaacs (HJBI, for short) 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 [20].
Therereallyexistsomesystemswhicharemodeledonlybystochasticsystemswhoseevolutionsdepend
on the past history of the states. Bases on this phenomenon, Ji and Yang [13] investigated a controlled
system governed by a functional forward-backward stochastic differential equation (FBSDE, for short)
and proved the value function is the viscosity solution of the related path-dependent HJB equation.
In this paper, inspired by [1] and [13], we will investigatethe stochastic differentialgame problems of
the functional FBSDEs. Precisely, the dynamics of the stochastic differentialgames are describedby the
following functional SDE:
 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],
 Xγt,u,v = γ . (1.1)
t t

Andthe costfunctionalJ(γt;u,v)isdefinedasYγt;u,v(t)whichisthesolutionofthefollowingfunctional
BSDE:
 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),
 Yγt,u,v(T) = Φ(Xγt;u,v), s∈[t,T], (1.2)
T

where γ is a path on [0,t]. The driver f and Φ can be interpreted as the running cost and the terminal
t
cost, respectively. Also, they depend on the past history of the dynamics. (1.1) and (1.2) compose a
decoupled functional FBSDE. The concrete conditions on b,σ,f,Φ are shown in the later section.
Inthecontext,weadoptthestrategyagainstcontrolform. ThecostfunctionalJ(γ ;u,v)isexplained
t
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
W(γ ):= essinf esssupJ(γ ;u,β(u)), (1.3)
t t
β∈Bt,T u∈Ut,T
U(γ ):=esssupessinfJ(γ ;α(v),v) (1.4)
t t
α∈At,T v∈Vt,T
are the viscosity solutions of the following path-dependent HJBI equations, respectively:
D W(γ )+sup inf H(γ ,W,D W,D W,u,v)=0,
 t t t x xx
 u∈Uv∈V (1.5)
W(γ )=Φ(γ ), γ ∈Λ,
T T T

and
D U(γ )+ inf supH(γ ,U,D U,D U,u,v)=0,
 t t t x xx
 v∈V u∈U (1.6)
U(γ )=Φ(γ ), γ ∈Λ,
t T T

2
where
1
H(γ ,y,p,X,u,v)= tr(σσT(γ ,u,v)X)+p.b(γ ,u,v)+f(γ ,y,p.σ(γ ,u,v),u,v),
t t t t t
2
where (γ ,y,p,X)∈Λ×R×Rd×Sd (Sd denotes the set of d×d symmetric matrices).
t
To solve the above stochastic differential game problem, we need the functional Itô calculus and
path-dependent PDEs which are recently introduced by Dupire [7] (for a recent account of this theory,
the reader may consult [2, 3, 4]. And under the framework of functional Itô’s calculus, for the non-
MarkovianBSDEs, Peng andWang [22]deriveda nonlinear Feynman-Kacformula for classicalsolutions
of path-dependent PDEs. For the further development, the readers may refer to [8, 21] ).
In this paper, we apply the Girsanov transformation method in Buckdahn and Li [1] to prove the
determinacy of the value functions, which is different from the method introduced by Peng [19, 20].
Making use of this method and the functional Itô’s calculus (introducedby Dupire [7], anddeveloped by
Cont, Fournié [2, 3, 4]), we complete the study of the zero-sum two-player stochastic differential games
in the non-Markoviancase and present the lower and upper value functions of our stochastic differential
game are the viscosity solution of the corresponding path-dependent HJBI equations, respectively.
Different from the HJBI equations derived for stochastic delay systems, we establish the dynamic
programming principle and derive the HJBI equation in the new framework of functional Itô calculus.
This paper is organized as follows: Section 2 recalls the functional Itˆo calculus and the well-known
results of BSDEs we will use later. In Section 3, we formulate our stochastic differential games and get
the correspondingDPP. Basedonthe obtainedDPP, inSection4 wederivethe mainresultofthe 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 [7]. Here we
follow the notations in [7].
Let T >0 be fixed. For each t∈[0,T], we denote Λ the set of càdlàg functions from [0,t] to Rd.
t
For γ ∈ Λ , denote γ(s) by the value of γ at time s ∈ [0,T]. Thus γ = (γ(s)) is a càdlàg
T 0≤s≤T
process on [0,T] and its value at time s is γ(s). γ =(γ(s)) ∈Λ is the path of γ up to time t. We
t 0≤s≤t t
denote Λ = Λ . For each γ ∈ Λ and x ∈ Rd, γ (s) is denoted by the value of γ at s ∈ [0,t] and
t t t t
t∈S[0,T]
γx :=(γ (s) ,γ (t)+x) which is also an element in Λ .
t t 0≤s<t t t
We now introduce a distance on Λ. Let h·,·i and |·| denote the inner product and norm in Rd. For
each 0≤t,t¯≤T and γt,γt¯∈Λ, we set
kγ k:= sup |γ (s)|,
t t
s∈[0,t]
kγt−γ¯t¯k:= sup |γt(s∧t)−γ¯t¯(s∧t¯)|,
s∈[0,t∨t¯]
d∞(γt,γ¯t¯):= sup |γt(s∧t)−γ¯t¯(s∧t¯)|+|t−t¯|.
s∈[0,t∨t¯]
It is obvious that Λ is a Banach space with respect to k·k. Since Λ is not a linear space, d is not a
t ∞
norm.
3
Definition 2.1. A functional u : Λ 7→ R is Λ-continuous at γ ∈ Λ, if for any ε > 0 there exists δ > 0
t
such that for each γ¯t¯∈Λ with d∞(γt,γ¯t¯)<δ, we have |u(γt)−u(γ¯t¯)|<ε.
u is said to be Λ-continuous if it is Λ-continuous at each γ ∈Λ.
t
Definition 2.2. Let v :Λ7→R and γ ∈Λ be given. If there exists p∈Rd, such that
t
v(γx)=v(γ )+hp,xi+o(|x|),as x→0,x∈Rd.
t t
Then we say that v is (vertically) differentiable at t and denote the gradient of D v(γ )=p. v is said to
x t
be vertically differentiable in Λ, if D v(γ ) exists for each γ ∈ Λ. The Hessian D v(γ ) can be defined
x t t xx t
similarly. It is an S(d)-valued function defined on Λ, where S(d) is the space of all d×d symmetric
matrices.
For each γ ∈Λ, we denote γ (r):=γ (r)1 (r)+γ (t)1 (r), r∈[0,s]. It is clear that γ ∈Λ .
t t,s t [0,t) t [t,s] t,s s
Definition 2.3. For a given γ ∈Λ, if we have
t
v(γ )=v(γ )+a(s−t)+o(|s−t|), as s→t, s≥t,
t,s t
then we say that v(γ ) is (horizontally) differentiable in t at γ and denote D v(γ ) = a. v is said to be
t t t t
horizontally differentiable in Λ if D v(γ ) exists for each γ ∈Λ.
t t t
Definition 2.4. Define Cj,k(Λ) as the set of function v := (v(γ )) defined on Λ which are j times
t γt∈Λ
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 [7] and then
developed by Cont and Fournié [4] for a more general formulation.
Theorem 2.1. (Functional Itô’s formula). Let (Ω,F,(F ) ,P) be a probability space, if X is a
t t∈[0,T]
continuous semi-martingale and v is in C1,2(Λ), then for any t∈[0,T),
t t 1 t
v(X )−v(X )= D v(X )ds+ D v(X )dX(s)+ D v(X )dhXi(s), P-a.s.
t 0 Z s s Z x s 2Z xx s
0 0 0
2.2 BSDEs
In this section, we recollect some important results which will be used in our stochastic differential
game problems.
Let(Ω,F,P)betheWienerspace,whereΩisthesetofcontinuousfunctionsfrom[0,T]toRd starting
from 0 (Ω = C ([0,T];Rd)), F is the completed Borel σ-algebra over Ω, and P is the Wiener measure.
0
Let B be the canonical process: B(ω,s) = ω ,s ∈ [0,T], ω ∈ Ω. We denote by F = {F , 0 ≤ s ≤ T}
s s
the natural filtration generated by {B(t)} and augmented by all P-null sets, i.e., F = σ{B(r),r ≤
t≥0 s
s}∨N , s∈[0,T]; Fs =σ(B(r)−B(t),t≤r≤s)∨N , where N is the set of all P-null subsets and
P t P P
T is a fixed real time horizon. We introduce the following two spaces of processes which will be used
frequently:
S2(0,T;Rn):={(ψ(t)) Rn-valued F-adapted continuous process:
0≤t≤T
E[ sup |ψ(t)|2]<+∞};
0≤t≤T
H2(0,T;Rn):={(ψ(t)) Rn-valued F-progressivelymeasurable process:
0≤t≤T
kψ k2=E[ T |ψ(t)|2dt]<+∞}.
0
R
Now we consider a function g : Ω×[0,T]×R×Rd → R, such that (g(t,y,z)) is progressively
t∈[0,T]
measurable for each (y,z) in R×Rd, and satisfies the following assumptions throughout the paper:
4
(A1) There exists a constant C ≥0 such that, P-a.s., for all t∈[0,T], y ,y ∈R, z ,z ∈Rd,
1 2 1 2
|g(t,y ,z )−g(t,y ,z )|≤C(|y −y |+|z −z |),
1 1 2 2 1 2 1 2
(A2) g(·,0,0)∈H2(0,T;R).
The following results on BSDEs are well-known. The readers may refer to Pardoux and Peng [14].
Lemma 2.1. Under the assumptions (A1) and (A2), for any random variable ξ ∈ L2(Ω,F ,P), the
T
BSDE
T T
y(t)=ξ+ g(s,y(s),z(s))ds− z(s)dB(s), 0≤t≤T (2.1)
Z Z
t t
has a unique adapted solution
(y(t),z(t)) ∈S2(0,T;R)×H2(0,T;Rd).
t∈[0,T]
In the following, we always suppose the driving coefficient g of a BSDE satisfies (A1) and (A2).
Lemma 2.2. (Comparison Theorem) Given two coefficients g , g satisfying (A1) and (A2) and two
1 2
terminal values ξ ,ξ ∈L2(Ω,F ,P), we denote by (y1,z1) and (y2,z2) the solution of a BSDE with the
1 2 T
data (ξ ,g ) and (ξ ,g ), respectively. Then we have :
1 1 2 2
(i) If ξ ≥ξ and g ≥g , P-a.s., then y1(t)≥y2(t), P-a.s., for all t∈[0,T].
1 2 1 2
(ii) (Strict monotonicity) If, in addition to (i), we also assume that P(ξ >ξ )>0, then P(y1(t)>
1 2
y2(t))>0, 0≤t≤T, and, in particular, y1(0)>y2(0).
With the notations in Lemma 2.2, we assume that, for some g :Ω×[0,T]×R×Rd −→R satisfying
(A1) and (A2), the drivers g have the following form:
i
g (s,yi(s),zi(s))=g(s,yi(s),zi(s))+ϕ (s), dsdP-a.e., i=1,2,
i i
whereϕ ∈H2(0,T;R).Then,forallterminalvaluesξ , ξ ∈L2(Ω,F ,P),wehavethefollowingresults.
i 1 2 T
Lemma 2.3. The difference of the solutions (y1,z1) and (y2,z2) of BSDE (2.1) with the data (ξ ,g )
1 1
and (ξ ,g ), respectively, satisfies the following estimate:
2 2
|y1(t)−y2(t)|2+ 1E[ T eβ(s−t)(|y1(s)−y2(s)|2+|z1(s)−z2(s)|2)ds|F ]
2 t t
R
≤E[eβ(T−t)|ξ −ξ |2|F ]+E[ T eβ(s−t)|ϕ (s)−ϕ (s)|2ds|F ],
1 2 t t 1 2 t
R
P-a.s., for all 0≤t≤T, where β =16(1+C2).
Proof. For the proof the reader is referred to Proposition 2.1 in El Karoui, Peng, and Quenez [12] or
Theorem 2.3 in Peng [20].
3 A DPP for Stochastic Differential Games of functional FBSDEs
In this section, we consider the stochastic differential games of functional FBSDEs.
First we introduce the background of stochastic differential games. Suppose that the control state
spaces U, 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.
5
Let us give the following mappings
b:Λ×U ×V −→Rn,
σ :Λ×U ×V −→Rn×d,
f :Λ×R×Rd×U ×V −→R.
For given admissible controls u(·) ∈ U, v(·) ∈ V, and t ∈ [0,T], γ ∈ Λ, we consider the following
t
functional forward-backwardstochastic system
dXγt;u,v(s) = b(Xγt;u,v,u(s),v(s))ds+σ(Xγt;u,v,u(s),v(s))dB(s), s∈[t,T],
 s s
 XdYγtγ,tu;u,v,v(s) == γ−f,(Xsγt;u,v,Yγt;u,v(s),Zγt;u,v(s),u(s),v(s))ds+Zγt;u,v(s)dB(s), (3.1)
t t
 Yγt,u,v(T) = Φ(XTγt;u,v),
Now, we presentthe assumptions to ensure the existence anduniqueness of the solution offunctional
FBSDEs (3.1).
(H) (i)Forallt∈[0,T],u∈U, v ∈V,x ∈Λ, y ∈R, z ∈Rd,b(x ,u,v), σ(x ,u,v),andf(x ,y,z,u,v)
t t t t
are F -measurable.
t
(ii) There exists a constant C >0, such that, for all t∈[0,T], u∈U, v ∈V, for any x1,x2 ∈Λ,
t t
|b(x1,u,v)−b(x2,u,v)|≤C kx1−x2 k;
t t t t
|b(x ,u,v)|≤C(1+kx k), for any x ∈Λ.
t t t
(iii) There exists a constant C > 0, such that for all t ∈ [0,T], u ∈ U, v ∈ V, for any x1,x2 ∈
t t
Λ, y1,y2 ∈R, z1,z2 ∈Rd
|f(x1,y1,z1,u,v)−f(x2,y2,z2,u,v)|≤C(kx1−x2 k+|y1−y2|+|z1−z2|);
t t t t
|Φ(x1)−Φ(x2)|≤C kx1 −x2 k;
T T T T
and
|f(x ,0,0,u,v)|≤C(1+kx k);
t t
|Φ(x )|≤C(1+kx k), for any x ∈Λ.
T T t
Theorem 3.1. 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 (3.1).
We introduce the subspaces of admissible controls and the definitions of admissible strategies, which
are similar to [1].
Definition 3.1. An admissible control process u= (u ) (resp., v =(v ) ) for Player I (resp.,
r r∈[t,s] r r∈[t,s]
II) on [t,s] is an F -progressively measurable, U (resp., V)-valued process. The set of all admissible
r
controls for Player I (resp., II) on [t,s] is denote by U (resp., V ). If P{u≡u¯, a.e., in [t,s]}=1, we
t,s t,s
will identify both processes u and u¯ in U . Similarly we interpret v ≡v¯ on [t,s] in V .
t,s t,s
6
Definition3.2. AnonanticipativestrategyforPlayerIon[t,s](t<s≤T)isamappingα: V →U
t,s t,s
such that, for any F-stopping time S : Ω → [t,s] and any v ,v ∈ V , with v ≡ v on [[t,S]], it holds
1 2 t,s 1 2
that α(v ) ≡ α(v ) on [[t,S]]. Nonanticipative strategies for Player II on [t,s], β : U → V , are
1 2 t,s t,s
defined similarly. The set of all nonanticipative strategies α: V →U for Player I on [t,s] is denoted
t,s t,s
by A . The set of all nonanticipative strategies β : U →V for Player II on [t,s] is denoted by B .
t,s t,s t,s t,s
(Recall that [[t,S]]={(r,ω)∈[0,T]×Ω,t≤r≤S(ω).)
For given processes u(·) ∈ U , v(·) ∈ V , initial data t ∈ [0,T], γ ∈ Λ, the cost functional is
t,T t,T t
defined as follows:
J(γ ;u,v):=Yγt;u,v(t), γ ∈Λ, (3.2)
t t
where the process Yγt;u,v is defined by functional FBSDE (3.1).
For γ ∈Λ, we define the lower value function of our stochastic differential games
t
W(γ ):= essinf esssupJ(γ ;u,β(u)), (3.3)
t t
β∈Bt,T u∈Ut,T
and its upper value function
U(γ ):=esssupessinfJ(γ ;α(v),v). (3.4)
t t
α∈At,T v∈Vt,T
In Ji, Yang [13], they proved the following estimates:
Lemma 3.1. Under the assumption (H), there exists some constant C > 0 such that, for any t ∈
[0,T], γ ,γ¯ ∈Λ, u(·)∈U, v(·)∈V,
t t
E[ sup |Xγt,u,v(s)|2|Ft]≤C(1+kγtk2),
s∈[t,T]
E[ sup |Xγt,u,v(s)−Xγ¯t,u,v(s)|2|Ft]≤Ckγt−γ¯tk2,
s∈[t,T]
E[ sup |Yγt,u,v(s)|2+ tT |Zγt,u,v(s)|2ds|Ft]≤C(1+kγtk2),
s∈[t,T] R
E[ sup |Yγt,u,v(s)−Yγ¯t,u,v(s)|2+ tT |Zγt,u,v(s)−Zγ¯t,u,v(s)|2ds|Ft]≤Ckγt−γ¯tk2.
s∈[t,T] R
As we known, the essentialinfimum andessentialsupremum on a family of randomvariables are still
random variables. But by applying the method introduced by Buckdahn and Li [1], we get W(γ ) and
t
U(γ ) are deterministic.
t
Proposition 3.1. For any t ∈ [0,T], γ ∈ Λ, W(γ ) is a deterministic function in the sense that
t t
W(γ )=E[W(γ )], P-a.s.
t t
Proof. Let H denote the Cameron-Martin space of all absolutely continuous elements h ∈ Ω whose
derivative h˙ belongs to L2([0,T];Rd).
For any h ∈ H, we define the mapping τ ω := ω +h,ω ∈ Ω. It is easy to check that τ : Ω → Ω
h h
is a bijection, and its law is given by P ◦[τ ]−1 = exp{ T h˙(s)dB(s)− 1 T |h˙(s)|2ds}P. For any fixed
h 0 2 0
R R
t∈[0,T], set H ={h∈H|h(·)=h(·∧t)}. The proof can be separated into the following steps:
t
(1). For all u∈U , h∈H , J(γ ;u,v)(τ )=J(γ ;u(τ ),v(τ )), P-a.s.
t,T t t h t h h
7
First, we make the transformationfor the functional SDE:
 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)+RRtsb(Xrγt;u,v+(τhts),σu((Xτhrγ)t(;ur,)v,(vτ(hτ)Rh,)u((rτ)h))d(rr),v(τh)(r))dB(r),
R
 Xγt;u(τh),v(τh)(s) = γt(t)+Rtsb(Xrγt;u(τ+h)R,vts(τσh()X,urγ(tτ;uh()τ(hr)),v,(vτ(hτ)h,)u((rτ)h))d(rr),v(τh)(r))dB(r), (3.5)
then, from the uniqueness of the solution of the functional SDE, we get
Xγt;u,v(s)(τh)=Xγt;u(τh),v(τh)(s), for any s∈[t,T], P-a.s.
Similarly, using the transformation to the BSDE in (3.1) and comparing the obtained equation with the
BSDE obtainedfrom (3.1) by replacing the transformedcontrolprocessu(τ ), v(τ ) for u, v, due to the
h h
uniqueness of the solution of the functional BSDE, we obtain
Yγt;u,v(s)(τh) = Yγt;u(τh),v(τh)(s), for any s∈[t,T], P-a.s.,
Zγt;u,v(s)(τh) = Zγt;u(τh),v(τh)(s), dsdP-a.e. on [0,T]×Ω.
Hence
J(γ ;u,v)(τ )=J(γ ;u(τ ),v(τ )), P-a.s.
t h t h h
(2). For β ∈B , h∈H , let βh(u):=β(u(τ ))(τ ), u∈U . Then, βh ∈B .
t,T t −h h t,T t,T
Obviously, βh maps U into V . Moreover, this mapping is nonanticipating. Indeed, let S : Ω →
t,T t,T
[t,T]beanF-stoppingtimeandu ,u ∈U ,withu ≡u on[[t,S]]. Then,obviously,u (τ )≡u (τ )
1 2 t,T 1 2 1 −h 2 −h
on [[t,S(τ )]]. Therefore,
−h
βh(u )=β(u (τ ))(τ )=β(u (τ ))(τ )=βh(u ) on [[t,S]].
1 1 −h h 2 −h h 2
(3). For any h∈H , and β ∈B , we have
t t,T
{esssupJ(γ ;u,β(u))}(τ )=esssup{J(γ ;u,β(u))(τ )}, P-a.s.
t h t h
u∈Ut,T u∈Ut,T
Infact,forconvenience,settingI(γ ;β):=esssupJ(γ ;u,β(u)), β ∈B ,wehaveI(γ ;β)≥J(γ ;u,β(u)).
t t t,T t t
u∈Ut,T
Then I(γ ;β)(τ )≥J(γ ;u,β(u))(τ ), P-a.s., for all u∈U . Therefore,
t h t h t,T
{esssupJ(γ ;u,β(u))}(τ )≥esssup{J(γ ;u,β(u))(τ )}, P-a.s.
t h t h
u∈Ut,T u∈Ut,T
Onthe otherhand,foranyrandomvariableξ whichsatisfiesξ ≥J(γ ;u,β(u))(τ ),wehaveξ(τ )≥
t h −h
J(γ ;u,β(u)), P-a.s., for all u∈U . So ξ(τ )≥I(γ ;β), P-a.s., i.e. ξ ≥I(γ ;β)(τ ), P-a.s. Thus,
t t,T −h t t h
J(γ ;u,β(u))(τ )≥{esssupJ(γ ;u,β(u))}(τ ), P-a.s., for any u∈U .
t h t h t,T
u∈Ut,T
Therefore,
esssup{J(γ ;u,β(u))(τ )}≥{esssupJ(γ ;u,β(u))}(τ ), P-a.s.
t h t h
u∈Ut,T u∈Ut,T
8
From above we get
{esssupJ(γ ;u,β(u))}(τ )=esssup{J(γ ;u,β(u))(τ )}, P-a.s.
t h t h
u∈Ut,T u∈Ut,T
(4). Under the Girsanov transformation τ , W(γ ) is invariant, i.e.,
h t
W(γ )(τ )=W(γ ), P-a.s., for any h∈H.
t h t
In fact, similarly to the third step, for all h∈H , we can prove
t
{essinf I(γ ;β)}(τ )= essinf{I(γ ;β)(τ )}, P-a.s.
t h t h
β∈Bt,T β∈Bt,T
From the first step and the third one, for all h∈H , we have
t
W(γ )(τ ) = {essinf esssupJ(γ ;u,β(u))}(τ ),
t h t h
β∈Bt,T u∈Ut,T
= essinf esssup{J(γ ;u,β(u))(τ )},
t h
β∈Bt,T u∈Ut,T
= essinf esssup{J(γ ;u(τ ),βh(u(τ )))},
t h h
β∈Bt,T u∈Ut,T
= essinf esssupJ(γ ;u,β(u)),
t
β∈Bt,T u∈Ut,T
= W(γ ), P-a.s.
t
In the latter equality we have used {u(τ ) | u(·) ∈ U } = U and {βh | β ∈ B } = B . Therefore,
h t,T t,T t,T t,T
for any h∈H , W(γ )(τ )=W(γ ), P-a.s., and since W(γ ) is F -measurable, we have this relation for
t t h t t t
all h∈H.
Combined with the following auxiliary lemma we can complete the proof.
Lemma 3.2. Let ζ be a random variable defined over our classical Wiener space (Ω,F ,P), such that
T
ζ(τ )=ζ, P-a.s., for any h∈H. Then ζ =Eζ, P-a.s.
h
Proof. Its proof can be found in Buckdahn and Li [1].
From the definition of W(γ ) and Lemma 3.1, we have the following property.
t
Lemma 3.3. There exists some constant C >0 such that, for all 0≤t≤T, γ ,γ¯ ∈Λ,
t t
(i) |W(γ )−W(γ¯ )|≤Ckγ −γ¯ k;
t t t t
(3.6)
(ii) |W(γ )|≤C(1+kγ k).
t t
Now we adopt Peng’s notion of stochastic backward semigroup to discuss a generalized DPP for
our stochastic differential game (3.1), (3.3). The notation of stochastic backward semigroup was first
introduced by Peng [20] to prove the DPP for stochastic control problems. First we define the family of
(backward) semigroups associated with FBSDE (3.1).
For given initial data γ , a number 0 < δ ≤ T −t, admissible control processes u(·) ∈ U , v(·) ∈
t t,t+δ
V , we put
t,t+δ
Gγt;u,v[η]:=Y˜γt;u,v(s), s∈[t,t+δ],
s,t+δ
9
where η ∈ L2(Ω,Ftt+δ,P;Rn), (Y˜γt;u,v(s),Z˜γt;u,v(s))t≤s≤t+δ is the solution of the following functional
FBSDE with the time horizon t+δ:
 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),
 (3.7)
Y˜γt;u,v(t+δ) = η, s∈[t,t+δ].

Also, we have
Gγt;u,v[Φ(Xγt;u,v)]=Gγt;u,v[Yγt;u,v(t+δ)].
t,T T t,t+δ
Theorem 3.2. Under the assumption (H), the lower value function W(γ ) satisfies the following DPP:
t
for any γ ∈Λ, δ >0,
t
W(γ )= essinf esssup Gγt;u,β(u)[W(Xγt;u,β(u))].
t t,t+δ t+δ
β∈Bt,t+δu∈Ut,t+δ
The proof is given in the Appendix.
4 Viscosity Solutions of Path-dependent HJBI equation
In this section, we will study the following path-dependent PDE:
 DtW(γt)+H−(γt,W,DxW,DxxW)=0,
 (4.1)
W(γ )=Φ(γ ), γ ∈Λ,
T T T

and
 DtU(γt)+H+(γt,U,DxU,DxxU)=0,
 (4.2)
U(γ )=Φ(γ ), γ ∈Λ,
t T T
where 
H−(γ ,W,D W,D W)= sup inf H(γ ,W,D W,D W,u,v),
t x xx t x xx
u∈Uv∈V
H+(γ ,U,D U,D U)= inf supH(γ ,U,D U,D U,u,v),
t x xx t x xx
v∈V u∈U
and
H(γ ,y,p,X,u,v)= 1tr(σσT(γ ,u,v)X)+p.b(γ ,u,v)+f(γ ,y,p.σ(γ ,u,v),u,v),
t 2 t t t t
where (γ ,y,p,X)∈Λ×R×Rd×Sd (Sd denotes the set of d×d symmetric matrices).
t
We will show that the value function W(γ ) (resp., U(γ )) defined in (3.3) (resp., (3.4)) is a viscosity
t t
solutionofthecorrespondingequation(4.1)(resp.,(4.2)). Firstwegivethedefinitionofviscositysolution
forthiskindofPDEs. Formoreinformationonviscositysolution,thereaderisreferredtoCrandall,Ishii
and Lions [5].
Definition 4.1. A real-valued Λ-continuous function W ∈C(Λ) is called
(i) a viscosity subsolution of equation (4.1) if for any δ > 0, Γ ∈ C1,2(Λ), γ ∈ Λ satisfying Γ ≥ W on
l,b t
Λ and Γ(γ )=W(γ ), we have
t+s t t
0≤Ss≤δ
D Γ(γ )+H−(γ ,Γ,D Γ,D Γ)≥0,
t t t x xx
(ii) a viscosity supersolution of equation (4.1) if for any δ >0, Γ∈C1,2(Λ), γ ∈Λ satisfying Γ≤W on
l,b t
Λ and Γ(γ )=W(γ ), we have
t+s t t
0≤Ss≤δ
D Γ(γ )+H−(γ ,Γ,D Γ,D Γ)≤0,
t t t x xx
10