# Stochastic Optimization Theory of Backward Stochastic Differential Equations Driven by G-Brownian Motion

Authors: Zhonghao Zheng; Xiuchun Bi; Shuguang Zhang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in Hu et al. (2012), we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in Zhang (2011). Then we obtain a generalized dynamic programming principle, and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.


---

## Body

## 1. Introduction

Nonlinear BSDEs in the framework of linear expectation were introduced by Pardoux and Peng [1] in 1990. Then, a lot of researches were studied by many authors, and they provided various applications of BSDEs in stochastic control, finance, stochastic differential games, and second-order partial differential equations theory; see [2–9].The notion of sublinear expectation space was introduced by Peng [10–12], which is a generalization of classical probability space. The G-expectation, a type of sublinear expectation, has played an important role in the researches of sublinear expectation space recently. It can be regarded as a counterpart of the Wiener probability space in the linear case. Within this G-expectation framework, the G-Brownian motion is the canonical process. Besides, the notions of the G-martingales and the Itô integral with respect to G-Brownian motion were also derived. There are some new structures in these notions and some new applications in the financial models with volatility uncertainty; see Peng [12, 13].In the G-expectation framework, thanks to a series of studies [14–17], the complete representation theorem for G-martingales has been obtained by Peng et al. [18]. Due to this contribution, a natural formulation of BSDEs driven by G-Brownian motion was found by Hu et al. [19]. In addition, the existence and uniqueness of the solution to the BSDEs driven by G-Brownian motion have been proved. They also have given the comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion in [20]. So the complete theory of BSDEs driven by G-Brownian motion has been established.An important application of BSDEs is that we can define the recursive utility functions from BSDEs, which can index scaling risks in the study of economics and finance [21–24]. Based on these results, a type of significant stochastic optimal control problems under linear expectation with a BSDE as cost function was studied [2, 4, 7–9]. Under G-expectation, the similar problems will be useful in the future studies of finance models with volatility uncertainty. So we arise a natural question: can we construct the similar results in G-expectation framework? In [25, 26], Zhang have given the study about the stochastic control problems under G-expectation based on the preliminary theory of BSDEs driven by G-Brownian motion. When the complete results about BSDEs driven by G-Brownian motion were established in [19, 20], we tried to prove the complete results of stochastic optimization theory of BSDEs driven by G-Brownian motion in this paper.In this paper, we investigate the stochastic optimal control problems with a BSDE driven by G-Brownian motion constructed in [19, 20] as cost function. Based on the results in [19, 20], we obtain the dynamic programming principle under G-expectation. Besides, the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.The rest of the paper is organized as follows. In Section2, we recall the G-expectation framework and adapt it according to our objective. Besides, we give the related properties of forward and backward stochastic differential equations driven by G-Brownian motion, which will be needed in the sequel sections. In Section 3, the stochastic optimal control problems with a BSDE driven by G-Brownian motion as cost function are investigated and a dynamic programming principle under G-expectation is obtained. In Section 4, The value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.


## 2. Preliminaries

In this section, we recall the G-expectation framework established by Peng [10–12, 27]. Besides, we give some results about forward and backward stochastic differential equations driven by G-Brownian motion, which we need in the following sections. Some details can be found in [19, 20].


### 2.1. G-Expectation and G-Martingales

Definition 1.
LetΩ be a given set, and let ℋ be a linear space of real valued functions defined on Ω; namely, c∈ℋ for each constant c and |X|∈ℋ if X∈ℋ. The space ℋ can be considered as the space of random variables. A sublinear expectation 𝔼 is a functional 𝔼:ℋ→ℝ satisfying the following properties: for all X,Y∈ℋ, we have(i)
monotonicity:𝔼[X]≥𝔼[Y] if X≥Y;(ii)
constant preservation:𝔼[c]=c, for c∈ℝ;(iii)
subadditivity:𝔼[X+Y]≤𝔼[X]+𝔼[Y];(iv)
positive homogeneity:𝔼[λX]=λ𝔼[X], for λ≥0.
The triple(Ω,ℋ,𝔼) is called a sublinear expectation space.Definition 2 (G-normal distribution).
Ad-dimensional random vector X=(X1,…,Xd) on a sublinear expectation space (Ω,ℋ,𝔼) is called G-normally distributed if for each a,b≥0, we have
(1)X+bX¯=da2+b2X,
where X¯ is an independent copy of X; that is, X¯ and X are identically distributed, and X¯ is independent of X. Here, the letter G denotes the function
(2)G(A):=12𝔼[(AX,X)]:𝕊d↦ℝ,
where 𝕊d denotes the collection of all d×d symmetric matrices.Proposition 3.
LetX be G-normal distributed. The distribution of X is characterized by
(3)u(t,x)=𝔼[φ(x+tX)],φ∈Cb,Lip(ℝd).
In particular, 𝔼[φ(X)]=u(1,0), where u is the unique viscosity solution of the following parabolic PDE defined on [0,∞)×ℝd:
(4)∂tu-G(D2u)=0,u|t=0=φ,
where G is defined by (2).Remark 4.
It is easy to check thatG is a monotonic sublinear function defined on 𝕊(d) and G(A):=(1/2)𝔼[(AX,X)]≤(1/2)|A|𝔼[|X|2]=(1/2)|A|σ¯2 implies that there exists a bounded, convex, and closed subset Γ⊂𝕊d+ such that
(5)G(A)=12supγ∈ΓTr(γA),
where 𝕊d+ denotes the collection of nonnegative elements in 𝕊d. If there exist some β>0 such that G(A)-G(B)≥βtr[A-B] for any A≥B, we call the G-normal distribution nondegenerate, which is the case we consider throughout this paper.Definition 5.
LetΩ=C0d([0,T]), that is, the space of all ℝd-valued continuous paths (ωt)t∈[0,T] with ω0=0. The corresponding canonical process is Bt(ω)=ωt, t∈[0,T]. P0 is wiener measure. 𝔽={ℱtB}t≥0 is the filtration generated by B. We let ℋ:=Lip(ΩT) be a linear space of random variables for each fixed T≥0, where Lip(ΩT):={φ(Bt1,…,Btn):n≥1,t1,…,tn∈[0,T],φ∈Cb,Lip(ℝd×n)}.
(
i
) The G-expectation E^ is a sublinear expectation defined by
(6)E^[X]:=E~[φ(t1-t0ξ1,…,tn-tn-1ξn)],
for each X=φ(Bt1-Bt0,Bt2-Bt1,…,Btn-Btn-1), where (ξi)i=1n are identically distributed d-dimensional G-normally distributed random vectors in a sublinear expectation space (Ω~,ℋ~,E~) such that ξi+1 is independent of (ξ1,…,ξi) for each i=1,2,…,n-1. (Ω,ℋ,E^) is called G-expectation space and the canonical process {Bt}t∈[0,T] in the sublinear space (Ω,ℋ,E^) is called a G-Brownian motion.
(
ii
) The conditional G-expectation E^t of X∈Lip(ΩT) is defined by
(7)E^tj[φ(Bt1-Bt0,Bt2-Bt1,…,Btn-Btn-1)]:=ψ(Bt1-Bt0,…,Btj-Btj-1),
where ψ(x1,…,xj) = E~[φ(x1,…,xj, tj+1-tjξj+1,…, tn-tn-1ξn)].We denote byLGp(ΩT), p≥1, the completion of G-expectation space Lip(ΩT) under the norm ∥X∥p,G:=(E^[|X|p])1/p. For all t∈[0,T], E^[·] and E^t[·] are continuous mappings on Lip(ΩT) endowed with the norm ∥·∥1,G. Therefore, it can be extended continuously to LGp(ΩT).Definition 6.
A process{Mt}t≥0 is called a G-martingale if for each t∈[0,T], Mt∈LG1(Ωt), and for each s∈[0,t], we have E^s[Mt]=Ms.Now we introduce the Itô integral and quadratic variation process with respect to G-Brownian motion in G-expectation space.Definition 7.
Letp≥1 be fixed. For a given partition πT={t0,…,tN} of [0,T], we denote MGp,0(0,T) as the collection of the following type of simple processes:
(8)ηt(ω)=∑k=0N-1ξk(ω)1[tk,tk+1)(t),
where ξk∈Lip(Ωtk), k=0,1,2,…,N-1. We denote by MGp(0,T) the completion of MGp,0(0,T) under the norm ∥·∥MGp(0,T):={E^[∫0T‍|·|pdt]}1/p.Definition 8.
For eachη∈MG2,0(0,T), we define
(9)I(η)=∫0T‍ηtdBt:=∑j=0N-1ξj(Btj+1-Btj).
The mapping I:MG2,0(0,T)→LG2(ΩT) is continuous and thus can be continuously extended to MG2(0,T).Definition 9.
The quadratic variation process of G-Brownian motion is defined by(10)〈B〉t:=Bt2-2∫0t‍BsdBs,
which is a continuous, nondecreasing process.Definition 10.
We now define the integral of a processη∈MG1(0,T) with respect to 〈B〉 as follows:
(11)Q0,T(η)=∫0T‍ηtd〈B〉t≔∑j=0N-1ξj(〈B〉tj+1-〈B〉tj):MG1,0(0,T)⟶LG1(ΩT).
The mapping is continuous and can be extended to MG1(0,T) uniquely.Then, we detail some results about the quasianalysis theory constructed in [27].Theorem 11.
There exists a weakly compact probability measures family𝒫 on (Ω,ℬ(Ω)) such that
(12)E^[X]=supP∈𝒫Ep[X],∀X∈Lip(ΩT),𝒫 is called a set of probability measures that represents E^.Definition 12.
We define the capacity associated to𝒫, which is a weakly compact family of probability measure that represents E^, as follows:
(13)c^(A):=supP∈𝒫P(A),A∈ℬ(ΩT);c^ is also called the capacity induced by E^.Let(Ω0,ℱ0={ℱt0},ℱ,P0) be a filtered probability space, and let {Wt} be a d-dimensional Brownian motion under P0. [27] proved that 𝒫M:={P0∘X-1∣Xt=∫0t‍hsdWs,h∈Lℱ02([0,T];Γ1/2)} represents G-expectation E^, where Γ1/2:={γ1/2∣γ∈Γ} and Γ is the set in the representation of G(·) of the formula (5).Definition 13.
(
i
) Let c^  be the capacity induced by E^. A set A⊂Ω is polar if c^(A)=0. A property holds “quasi-surely” (q.s. for short) if it holds outside a polar set.
(
ii
) Let X and Y be two random variables; we say that X is a version of Y if X=Y q.s.Let∥ψ∥p,G=[E^(|ψ|p)]1/p for ψ∈Cb(ΩT). The completion of Cb(ΩT) and Lip(ΩT) under ∥·∥p,G is the same, and we denote them by LGp(ΩT).


### 2.2. Forward and Backward Stochastic Differential Equations Driven by G-Brownian Motion

We consider the following stochastic differential equations driven byd dimensional G-Brownian motion (G-SDE):
(14)Xt=X0+∫0t‍b(s,Xs)ds+∑i,j=1d∫0t‍hij(s,Xs)d〈Bi,Bj〉s+∑j=1d∫0t‍σj(s,Xs)dBsj,
where t∈[0,T], the initial condition X0∈ℝn is a given constant, b,hij,σj are given functions satisfying b(·,x), and hij(·,x), σj(·,x)∈MG2(0,T;ℝn) for each x∈ℝn and the Lipschitz condition, that is, |ϕ(t,x)-ϕ(t,x′)|≤K|x-x′|, for each t∈[0,T], x, x′∈ℝn, ϕ=b, hij, and σj, respectively. The solution is a process X∈MG2(0,T;ℝn) satisfying the G-SDE (14).Theorem 14 (see [12]).
There exists a unique solutionX∈MG2(0,T;ℝn) of the stochastic differential equation (14).Now, we give the results about BSDEs driven by G-Brownian motion in the G-expectation space(ΩT,LG1(ΩT),E^) with ΩT=C0([0,T],ℝd) and σ¯2=E^[B12]≥-E^[-B12]=σ_2>0. We consider the following type of G-BSDEs (we always use Einstein convention):
(15)Yt=ξ+∫tT‍f(s,Ys,Zs)ds+∑i,j=1d∫tT‍gij(s,Ys,Zs)d〈Bi,Bj〉s-∫tT‍ZsdBs-(KT-Kt),
where f(t,ω,y,z), gij(t,ω,y,z):[0,T]×ΩT×ℝ×ℝd→ℝ satisfy the following properties: there exist some β>1 such that(H1)
for anyy, z, f(·,·,y,z), gij(·,·,y,z)∈MGβ(0,T);(H2)
for someL>0,
(16)|f(t,ω,y,z)-f(t,ω,y′,z′)|+∑i,j=1d|gij(t,ω,y,z)-gij(t,ω,y′,z′)|≤L(|y-y′|+|z-z′|).For simplicity, we denote by𝔊(0,T) the collection of processes (Y,Z,K) such that Y∈SGα(0,T), Z∈HGα(0,T); K is a decreasing G-martingale with K0=0 and KT∈LGα(ΩT). Here, SGα(0,T) is the completion of SG0(0,T)={h(t,Bt1∧t,…,Btn∧t):t1,…,tn∈[0,T],h∈Cb,Lip(ℝn+1)} under ∥·∥sGp={E^[supt∈[0,T]|ηt|p]}1/p, and HGp(0,T) is the completion of MG0(0,T) under ∥·∥HGP={E^[(∫0T‍|ηs|2ds)p/2]}1/p.Definition 15.
Letξ∈LGβ(ΩT) with β>1; f and gij satisfy (H1) and (H2). A triplet of processes (Y,Z,K) is called a solution of (15) if for some 1<α≤β the following properties hold:
(
a
)  (Y,Z,K)∈𝔊Gα(0,T);
(
b
)
(17)
Y
t
=
ξ
+
∫
t
T
‍
f
(
s
,
Y
s
,
Z
s
)
d
s
+
∑
i
,
j
=
1
d
∫
t
T
‍
g
i
j
(
s
,
Y
s
,
Z
s
)
d
〈
B
i
,
B
j
〉
s
-
∫
t
T
‍
Z
s
d
B
s
-
(
K
T
-
K
t
)
.Theorem 16 (see [19]).
Assume thatξ∈LGβ(ΩT) and f,g satisfy (H1) and (H2) for some β>1. Then, (15) has a unique solution (Y,Z,K). Moreover, for any 1<α<β, we have Y∈SGα(0,T), Z∈HGα(0,T;ℝd), and KT∈LGα(ΩT).We have the following estimates.Proposition 17 (see [19]).
Letξ∈LGβ(ΩT), and f, gij satisfy (H1) and (H2) for some β>1. For some 1<α<β, (Y,Z,K)∈𝔊Gα(0,T) is solution of equation (15). Then(i)
There exists a constantCα:=C(α,T,G,L)>0such that(18)|Yt|α≤CαE^t[|ξ|α+∫tT‍|hs0|αds],E^[(∫0T‍|Zs|2ds)α/2]≤Cα{(E^[(∫0T‍hs0ds)α])1/2E^[supt∈[0,T]|Yt|α]+(E^[supt∈[0,T]|Yt|α])1/2(E^[(∫0T‍hs0ds)α])1/2},E^[|KT|α]≤Cα{E^[supt∈[0,T]|Yt|α]+E^[(∫0T‍hs0ds)α]},wherehs0=|f(s,0,0)|+∑i,j=1d|gij(s,0,0)|.(ii)
For any givenα<α′<β, there exists a constantCα,α′depending onα,α′,T,G,Lsuch that(19)E^[supt∈[0,T]|Yt|α]≤Cα,α′{[supt∈[0,T]E^t[(∫0T‍hs0ds)α′]]E^[supt∈[0,T]E^t[|ξ|α]]+(E^[supt∈[0,T]E^t[(∫0T‍hs0ds)α′]])α/α′+E^[supt∈[0,T]E^t[(∫0T‍hs0ds)α′]]}.Proposition 18 (see [20]).
Letξi∈LGβ(ΩT), i=1,2, and fi, giji satisfy (H1) and (H2) for some β>1. For some 1<α<β, (Yi,Zi,Ki)∈𝔊Gα(0,T) are solutions of (15) corresponding to ξi, fi, and giji. Set Y^t=Yt1-Yt2, Z^t=Zt1-Zt2, and K^t=Kt1-Kt2. (i)
There exists a constantCα:=C(α,T,G,L)>0such that(20)|Y^t|α≤CαE^t[|ξ^|α+∫tT‍|h^s|αds],whereξ^=ξ1-ξ2, h^s=|f1(s,Ys2,Zs2)-f2(s,Ys2,Zs2)|+∑i,j=1d|gij1(s,Ys2,Zs2)-gij2(s,Ys2,Zs2)|.(ii)
For any givenα′withα<α′<β, there exists a constantCα,α′depending onα,α′,T,G,Lsuch that(21)E^[supt∈[0,T]|Y^t|α]≤Cα,α′{[supt∈[0,T]E^t[(∫0T‍h^sds)α′]]E^[supt∈[0,T]E^t[|ξ^|α]]+(E^[supt∈[0,T]E^t[(∫0T‍h^sds)α′]])α/α′+E^[supt∈[0,T]E^t[(∫0T‍h^sds)α′]]}.Theorem 19 (see [20]).
Let(Yti,Zti,Kti)t≤T, i=1,2, be the solutions of the following G-BSDEs:
(22)Yti=ξi+∫tT‍fi(s,Ysi,Zsi)ds+∫tT‍gi(s,Ysi,Zsi)d〈B〉s-∫tT‍ZsidBs-(KTi-Kti),
where ξi∈LGβ(ΩT), fi, gi satisfy (H1) and (H2) with β>1. If ξ1≥ξ2, f1≥f2, g1≥g2, then Yt1≥Yt2.Theorem 20 (see [20]).
Let(Yti,Zti,Kti)t≤T, i=1,2, be the solutions of the following G-BSDEs:
(23)Yti=ξi+∫tT‍fi(s,Ysi,Zsi)ds+∫tT‍gi(s,Ysi,Zsi)d〈B〉s-∫tT‍ZsidBs-(KTi-Kti)+VTi-Vti,
where ξi∈LGβ(ΩT), fi, gi satisfy (H1) and (H2), (Vti)t≤T are RCLL processes such that E^[supt∈[0,T]|Vti|β]<∞ with β>1. If ξ1≥ξ2, f1≥f2, g1≥g2, and Vt1-Vt2 is an increasing process, then Yt1≥Yt2.


## 2.1. G-Expectation and G-Martingales

Definition 1.
LetΩ be a given set, and let ℋ be a linear space of real valued functions defined on Ω; namely, c∈ℋ for each constant c and |X|∈ℋ if X∈ℋ. The space ℋ can be considered as the space of random variables. A sublinear expectation 𝔼 is a functional 𝔼:ℋ→ℝ satisfying the following properties: for all X,Y∈ℋ, we have(i)
monotonicity:𝔼[X]≥𝔼[Y] if X≥Y;(ii)
constant preservation:𝔼[c]=c, for c∈ℝ;(iii)
subadditivity:𝔼[X+Y]≤𝔼[X]+𝔼[Y];(iv)
positive homogeneity:𝔼[λX]=λ𝔼[X], for λ≥0.
The triple(Ω,ℋ,𝔼) is called a sublinear expectation space.Definition 2 (G-normal distribution).
Ad-dimensional random vector X=(X1,…,Xd) on a sublinear expectation space (Ω,ℋ,𝔼) is called G-normally distributed if for each a,b≥0, we have
(1)X+bX¯=da2+b2X,
where X¯ is an independent copy of X; that is, X¯ and X are identically distributed, and X¯ is independent of X. Here, the letter G denotes the function
(2)G(A):=12𝔼[(AX,X)]:𝕊d↦ℝ,
where 𝕊d denotes the collection of all d×d symmetric matrices.Proposition 3.
LetX be G-normal distributed. The distribution of X is characterized by
(3)u(t,x)=𝔼[φ(x+tX)],φ∈Cb,Lip(ℝd).
In particular, 𝔼[φ(X)]=u(1,0), where u is the unique viscosity solution of the following parabolic PDE defined on [0,∞)×ℝd:
(4)∂tu-G(D2u)=0,u|t=0=φ,
where G is defined by (2).Remark 4.
It is easy to check thatG is a monotonic sublinear function defined on 𝕊(d) and G(A):=(1/2)𝔼[(AX,X)]≤(1/2)|A|𝔼[|X|2]=(1/2)|A|σ¯2 implies that there exists a bounded, convex, and closed subset Γ⊂𝕊d+ such that
(5)G(A)=12supγ∈ΓTr(γA),
where 𝕊d+ denotes the collection of nonnegative elements in 𝕊d. If there exist some β>0 such that G(A)-G(B)≥βtr[A-B] for any A≥B, we call the G-normal distribution nondegenerate, which is the case we consider throughout this paper.Definition 5.
LetΩ=C0d([0,T]), that is, the space of all ℝd-valued continuous paths (ωt)t∈[0,T] with ω0=0. The corresponding canonical process is Bt(ω)=ωt, t∈[0,T]. P0 is wiener measure. 𝔽={ℱtB}t≥0 is the filtration generated by B. We let ℋ:=Lip(ΩT) be a linear space of random variables for each fixed T≥0, where Lip(ΩT):={φ(Bt1,…,Btn):n≥1,t1,…,tn∈[0,T],φ∈Cb,Lip(ℝd×n)}.
(
i
) The G-expectation E^ is a sublinear expectation defined by
(6)E^[X]:=E~[φ(t1-t0ξ1,…,tn-tn-1ξn)],
for each X=φ(Bt1-Bt0,Bt2-Bt1,…,Btn-Btn-1), where (ξi)i=1n are identically distributed d-dimensional G-normally distributed random vectors in a sublinear expectation space (Ω~,ℋ~,E~) such that ξi+1 is independent of (ξ1,…,ξi) for each i=1,2,…,n-1. (Ω,ℋ,E^) is called G-expectation space and the canonical process {Bt}t∈[0,T] in the sublinear space (Ω,ℋ,E^) is called a G-Brownian motion.
(
ii
) The conditional G-expectation E^t of X∈Lip(ΩT) is defined by
(7)E^tj[φ(Bt1-Bt0,Bt2-Bt1,…,Btn-Btn-1)]:=ψ(Bt1-Bt0,…,Btj-Btj-1),
where ψ(x1,…,xj) = E~[φ(x1,…,xj, tj+1-tjξj+1,…, tn-tn-1ξn)].We denote byLGp(ΩT), p≥1, the completion of G-expectation space Lip(ΩT) under the norm ∥X∥p,G:=(E^[|X|p])1/p. For all t∈[0,T], E^[·] and E^t[·] are continuous mappings on Lip(ΩT) endowed with the norm ∥·∥1,G. Therefore, it can be extended continuously to LGp(ΩT).Definition 6.
A process{Mt}t≥0 is called a G-martingale if for each t∈[0,T], Mt∈LG1(Ωt), and for each s∈[0,t], we have E^s[Mt]=Ms.Now we introduce the Itô integral and quadratic variation process with respect to G-Brownian motion in G-expectation space.Definition 7.
Letp≥1 be fixed. For a given partition πT={t0,…,tN} of [0,T], we denote MGp,0(0,T) as the collection of the following type of simple processes:
(8)ηt(ω)=∑k=0N-1ξk(ω)1[tk,tk+1)(t),
where ξk∈Lip(Ωtk), k=0,1,2,…,N-1. We denote by MGp(0,T) the completion of MGp,0(0,T) under the norm ∥·∥MGp(0,T):={E^[∫0T‍|·|pdt]}1/p.Definition 8.
For eachη∈MG2,0(0,T), we define
(9)I(η)=∫0T‍ηtdBt:=∑j=0N-1ξj(Btj+1-Btj).
The mapping I:MG2,0(0,T)→LG2(ΩT) is continuous and thus can be continuously extended to MG2(0,T).Definition 9.
The quadratic variation process of G-Brownian motion is defined by(10)〈B〉t:=Bt2-2∫0t‍BsdBs,
which is a continuous, nondecreasing process.Definition 10.
We now define the integral of a processη∈MG1(0,T) with respect to 〈B〉 as follows:
(11)Q0,T(η)=∫0T‍ηtd〈B〉t≔∑j=0N-1ξj(〈B〉tj+1-〈B〉tj):MG1,0(0,T)⟶LG1(ΩT).
The mapping is continuous and can be extended to MG1(0,T) uniquely.Then, we detail some results about the quasianalysis theory constructed in [27].Theorem 11.
There exists a weakly compact probability measures family𝒫 on (Ω,ℬ(Ω)) such that
(12)E^[X]=supP∈𝒫Ep[X],∀X∈Lip(ΩT),𝒫 is called a set of probability measures that represents E^.Definition 12.
We define the capacity associated to𝒫, which is a weakly compact family of probability measure that represents E^, as follows:
(13)c^(A):=supP∈𝒫P(A),A∈ℬ(ΩT);c^ is also called the capacity induced by E^.Let(Ω0,ℱ0={ℱt0},ℱ,P0) be a filtered probability space, and let {Wt} be a d-dimensional Brownian motion under P0. [27] proved that 𝒫M:={P0∘X-1∣Xt=∫0t‍hsdWs,h∈Lℱ02([0,T];Γ1/2)} represents G-expectation E^, where Γ1/2:={γ1/2∣γ∈Γ} and Γ is the set in the representation of G(·) of the formula (5).Definition 13.
(
i
) Let c^  be the capacity induced by E^. A set A⊂Ω is polar if c^(A)=0. A property holds “quasi-surely” (q.s. for short) if it holds outside a polar set.
(
ii
) Let X and Y be two random variables; we say that X is a version of Y if X=Y q.s.Let∥ψ∥p,G=[E^(|ψ|p)]1/p for ψ∈Cb(ΩT). The completion of Cb(ΩT) and Lip(ΩT) under ∥·∥p,G is the same, and we denote them by LGp(ΩT).


## 2.2. Forward and Backward Stochastic Differential Equations Driven by G-Brownian Motion

We consider the following stochastic differential equations driven byd dimensional G-Brownian motion (G-SDE):
(14)Xt=X0+∫0t‍b(s,Xs)ds+∑i,j=1d∫0t‍hij(s,Xs)d〈Bi,Bj〉s+∑j=1d∫0t‍σj(s,Xs)dBsj,
where t∈[0,T], the initial condition X0∈ℝn is a given constant, b,hij,σj are given functions satisfying b(·,x), and hij(·,x), σj(·,x)∈MG2(0,T;ℝn) for each x∈ℝn and the Lipschitz condition, that is, |ϕ(t,x)-ϕ(t,x′)|≤K|x-x′|, for each t∈[0,T], x, x′∈ℝn, ϕ=b, hij, and σj, respectively. The solution is a process X∈MG2(0,T;ℝn) satisfying the G-SDE (14).Theorem 14 (see [12]).
There exists a unique solutionX∈MG2(0,T;ℝn) of the stochastic differential equation (14).Now, we give the results about BSDEs driven by G-Brownian motion in the G-expectation space(ΩT,LG1(ΩT),E^) with ΩT=C0([0,T],ℝd) and σ¯2=E^[B12]≥-E^[-B12]=σ_2>0. We consider the following type of G-BSDEs (we always use Einstein convention):
(15)Yt=ξ+∫tT‍f(s,Ys,Zs)ds+∑i,j=1d∫tT‍gij(s,Ys,Zs)d〈Bi,Bj〉s-∫tT‍ZsdBs-(KT-Kt),
where f(t,ω,y,z), gij(t,ω,y,z):[0,T]×ΩT×ℝ×ℝd→ℝ satisfy the following properties: there exist some β>1 such that(H1)
for anyy, z, f(·,·,y,z), gij(·,·,y,z)∈MGβ(0,T);(H2)
for someL>0,
(16)|f(t,ω,y,z)-f(t,ω,y′,z′)|+∑i,j=1d|gij(t,ω,y,z)-gij(t,ω,y′,z′)|≤L(|y-y′|+|z-z′|).For simplicity, we denote by𝔊(0,T) the collection of processes (Y,Z,K) such that Y∈SGα(0,T), Z∈HGα(0,T); K is a decreasing G-martingale with K0=0 and KT∈LGα(ΩT). Here, SGα(0,T) is the completion of SG0(0,T)={h(t,Bt1∧t,…,Btn∧t):t1,…,tn∈[0,T],h∈Cb,Lip(ℝn+1)} under ∥·∥sGp={E^[supt∈[0,T]|ηt|p]}1/p, and HGp(0,T) is the completion of MG0(0,T) under ∥·∥HGP={E^[(∫0T‍|ηs|2ds)p/2]}1/p.Definition 15.
Letξ∈LGβ(ΩT) with β>1; f and gij satisfy (H1) and (H2). A triplet of processes (Y,Z,K) is called a solution of (15) if for some 1<α≤β the following properties hold:
(
a
)  (Y,Z,K)∈𝔊Gα(0,T);
(
b
)
(17)
Y
t
=
ξ
+
∫
t
T
‍
f
(
s
,
Y
s
,
Z
s
)
d
s
+
∑
i
,
j
=
1
d
∫
t
T
‍
g
i
j
(
s
,
Y
s
,
Z
s
)
d
〈
B
i
,
B
j
〉
s
-
∫
t
T
‍
Z
s
d
B
s
-
(
K
T
-
K
t
)
.Theorem 16 (see [19]).
Assume thatξ∈LGβ(ΩT) and f,g satisfy (H1) and (H2) for some β>1. Then, (15) has a unique solution (Y,Z,K). Moreover, for any 1<α<β, we have Y∈SGα(0,T), Z∈HGα(0,T;ℝd), and KT∈LGα(ΩT).We have the following estimates.Proposition 17 (see [19]).
Letξ∈LGβ(ΩT), and f, gij satisfy (H1) and (H2) for some β>1. For some 1<α<β, (Y,Z,K)∈𝔊Gα(0,T) is solution of equation (15). Then(i)
There exists a constantCα:=C(α,T,G,L)>0such that(18)|Yt|α≤CαE^t[|ξ|α+∫tT‍|hs0|αds],E^[(∫0T‍|Zs|2ds)α/2]≤Cα{(E^[(∫0T‍hs0ds)α])1/2E^[supt∈[0,T]|Yt|α]+(E^[supt∈[0,T]|Yt|α])1/2(E^[(∫0T‍hs0ds)α])1/2},E^[|KT|α]≤Cα{E^[supt∈[0,T]|Yt|α]+E^[(∫0T‍hs0ds)α]},wherehs0=|f(s,0,0)|+∑i,j=1d|gij(s,0,0)|.(ii)
For any givenα<α′<β, there exists a constantCα,α′depending onα,α′,T,G,Lsuch that(19)E^[supt∈[0,T]|Yt|α]≤Cα,α′{[supt∈[0,T]E^t[(∫0T‍hs0ds)α′]]E^[supt∈[0,T]E^t[|ξ|α]]+(E^[supt∈[0,T]E^t[(∫0T‍hs0ds)α′]])α/α′+E^[supt∈[0,T]E^t[(∫0T‍hs0ds)α′]]}.Proposition 18 (see [20]).
Letξi∈LGβ(ΩT), i=1,2, and fi, giji satisfy (H1) and (H2) for some β>1. For some 1<α<β, (Yi,Zi,Ki)∈𝔊Gα(0,T) are solutions of (15) corresponding to ξi, fi, and giji. Set Y^t=Yt1-Yt2, Z^t=Zt1-Zt2, and K^t=Kt1-Kt2. (i)
There exists a constantCα:=C(α,T,G,L)>0such that(20)|Y^t|α≤CαE^t[|ξ^|α+∫tT‍|h^s|αds],whereξ^=ξ1-ξ2, h^s=|f1(s,Ys2,Zs2)-f2(s,Ys2,Zs2)|+∑i,j=1d|gij1(s,Ys2,Zs2)-gij2(s,Ys2,Zs2)|.(ii)
For any givenα′withα<α′<β, there exists a constantCα,α′depending onα,α′,T,G,Lsuch that(21)E^[supt∈[0,T]|Y^t|α]≤Cα,α′{[supt∈[0,T]E^t[(∫0T‍h^sds)α′]]E^[supt∈[0,T]E^t[|ξ^|α]]+(E^[supt∈[0,T]E^t[(∫0T‍h^sds)α′]])α/α′+E^[supt∈[0,T]E^t[(∫0T‍h^sds)α′]]}.Theorem 19 (see [20]).
Let(Yti,Zti,Kti)t≤T, i=1,2, be the solutions of the following G-BSDEs:
(22)Yti=ξi+∫tT‍fi(s,Ysi,Zsi)ds+∫tT‍gi(s,Ysi,Zsi)d〈B〉s-∫tT‍ZsidBs-(KTi-Kti),
where ξi∈LGβ(ΩT), fi, gi satisfy (H1) and (H2) with β>1. If ξ1≥ξ2, f1≥f2, g1≥g2, then Yt1≥Yt2.Theorem 20 (see [20]).
Let(Yti,Zti,Kti)t≤T, i=1,2, be the solutions of the following G-BSDEs:
(23)Yti=ξi+∫tT‍fi(s,Ysi,Zsi)ds+∫tT‍gi(s,Ysi,Zsi)d〈B〉s-∫tT‍ZsidBs-(KTi-Kti)+VTi-Vti,
where ξi∈LGβ(ΩT), fi, gi satisfy (H1) and (H2), (Vti)t≤T are RCLL processes such that E^[supt∈[0,T]|Vti|β]<∞ with β>1. If ξ1≥ξ2, f1≥f2, g1≥g2, and Vt1-Vt2 is an increasing process, then Yt1≥Yt2.


## 3. A DPP for Stochastic Optimal Control Problems under G-Expectation

Now we introduce the setting for stochastic optimal control problems underG-expectation. We suppose that the control state space V is a compact metric space. Let the set of admissible control processes 𝒰 for the player be a set of V-valued stochastic processes in MG2([t,T];ℝn), t∈[0,T]. For a given admissible control υ(·)∈𝒰, the corresponding orbit, which regards t as the initial time and ξ∈LG2(Ωt;ℝn) as the initial state, is defined by the solution of the following type of G-SDE:
(24)dXst,ξ;υ=b(s,Xst,ξ;υ,υs)ds+∑i,j=1dhij(s,Xst,ξ;υ,υs)d〈Bi,Bj〉s+∑j=1dσj(s,Xst,ξ;υ,υs)dBsj,s∈[t,T],Xtt,ξ;υ=ξ,
where b, hij, σj:[0,T]×ℝn×𝒰→ℝn are deterministic functions and satisfy the following conditions (H3): (A1)
hij=hji for 1≤i,j≤d;(A2)
For every fixed(x,υ)∈ℝn×𝒰, b(·,x,υ), hij(·,x,υ), σj(·,x,υ) are continuous in t;(A3)
There exists a constantL>0, for any t∈[0,T], x, x′∈ℝn, υ, υ′∈𝒰 such that
(25)|b(t,x,υ)-b(t,x′,υ′)|+∑i,j=1d|hij(t,x,υ)-hij(t,x′,υ′)|+∑j=1d|σj(t,x,υ)-σj(t,x′,υ′)|≤L(|x-x′|+|υ-υ′|).From the assumption (H3), we can get global linear growth conditions forb, hij, σj; that is, there exists C>0 such that, for t∈[0,T], x∈ℝn, υ∈𝒰, |b(t,x,υ)|+∑i,j=1d|hij(t,x,υ)|+∑j=1d|σj(t,x,υ)|≤C(1+|x|+|υ|). Obviously, under the above assumptions, for any υ(·)∈𝒰, G-SDE (24) has a unique solution. Moreover, we have the following estimates.Proposition 21.
Letξ, ξ′∈LGp(Ωt;ℝn) with p≥2, υ(·), υ′(·)∈𝒰, t∈[0,T], and δ∈[0,T-t]; then we have
(26)E^t[|Xt+δt,ξ;υ-Xt+δt,ξ′;υ′|p]≤C(|ξ-ξ′|p+∫tt+δ‍E^t|υr-υr′|pdr),E^t[|Xt+δt,ξ;υ|p]≤C(1+|ξ|p),E^t[sups∈[t,t+δ]|Xst,ξ;υ-ξ|p]≤C(1+|ξ|p)δp/2,
where C depends on L,G,p,n,T.Proof.
The proof is similar to the proof of Proposition 4.1 in [20].Now we give bounded functionsΦ:ℝn→ℝ, f:[0,T]×ℝn×ℝ×ℝd×𝒰→ℝ, gij:[0,T]×ℝn×ℝ×ℝd×𝒰→ℝ that satisfy the following conditions: (H4) (i)
gij=gji for 1≤i,j≤d.(ii)
For every fixed(x,y,z,υ)∈ℝn×ℝ×ℝn×𝒰, f(·,x,y,z,υ) and gij(·,x,y,z,υ) are continuous in t, 1≤i,j≤d.(iii)
There exist a constantL>0, for t∈[0,T], x, x′∈ℝn, y, y′∈ℝ, z, z′∈ℝd, υ, υ′∈𝒰, such that
(27)|Φ(x)-Φ(x′)|≤L(|x-x′|),|f(t,x,y,z,υ)-f(t,x′,y′,z′,υ′)|+∑i,j=1d|gij(t,x,y,z,υ)-gij(t,x′,y′,z′,υ′)|≤L(|x-x′|+|y-y′|+|z-z′|+|υ-υ′|).From (H4), we have thatΦ, f, and gij also satisfy global linear growth condition in x; that is, there exists C>0, such that for all 0≤t≤T, υ∈𝒰, x∈ℝn,
(28)|Φ(x)|+|f(t,x,0,0,υ)|+|gij(t,x,0,0,υ)|≤C(1+|x|+|υ|).
For any υ∈𝒰 and ξ∈LG2(Ωt,ℝn), the mappings f(s,x,y,z,υ):=f(s,Xst,ξ;υ,y,z,υs) and gij(s,x,y,z,υ)=gij(s,Xst,ξ;υ,y,z,υs), where (s,y,z)∈[0,T]×ℝ×ℝd satisfy the conditions of Theorem 16 on the interval [t,T]. Therefore, there exists a unique solution for the following G-BSDE:
(29)Yst,ξ;υ=Φ(XTt,ξ;υ)+∫sT‍f(r,Xrt,ξ;υ,Yrt,ξ;υ,Zrt,ξ;υ,υr)dr-∫sT‍Zrt,ξ;υdBr-(KTt,ξ;υ-Kst,ξ;υ)+∑i,j=1d∫sT‍gij(r,Xrt,ξ;υ,Yrt,ξ;υ,Zrt,ξ;υ,υr)d〈Bi,Bj〉r,
where Xt,ξ;υ is introduced by (24).Proposition 22.
For eachξ, ξ′∈LGp(Ωt;ℝn) with p≥2 and υ(·), υ′(·)∈𝒰, we have
(30)|Ytt,ξ;υ-Ytt,ξ′;υ|≤C|ξ-ξ′|,|Ytt,ξ;υ|≤C(1+|ξ|),|Ytt,ξ;υ-Ytt,ξ;υ′|≤C(∫tT‍E^t|υ(r)-υ′(r)|2dr)1/2,
where C depends on L, G, n, and T.Proof.
The proof is similar to the Proposition 4.2 in [20].Given a control processυ(·)∈𝒰, we introduce an associated cost functional
(31)J(t,x;υ)=Ytt,x;υ,(t,x)∈[0,T]×ℝn,
where the process Ytt,ξ;υ is defined by G-BSDE (29). Similar to the proof of Theorem 4.4 in [20], we have that for t∈[0,T], ξ∈LG2(Ωt,ℝn),
(32)J(t,ξ;υ):=Ytt,ξ;υ.
But we are more interested in the case when ξ=x.Now we define the value function as follows:(33)u(t,x):=supυ(·)∈𝒰J(t,x;υ).Proposition 23.
u
(
t
,
x
) is a deterministic function of (t,x).Proof.
For a partition of[t,s]: t=t0<t1<⋯<tN=s, p≥2, t≤s≤T, we denote Lip(Ωst)∶={φ(Bt1-Bt,…,Btn-Bt):n≥1,t1,…,tn∈[t,s],φ∈Cb,Lip(ℝd×n)}, MGp,0,t(t,s;ℝn) by the collection of simple processes η(r)=∑k=0N-1ξk1[tk,tk+1)(r), where ξk∈Lip(Ωtkt;ℝn), k=0,1,2,…,N-1, and MGp,t(t,s;ℝn) by the completion of MGp,0,t(t,s;ℝn) under the norm ∥η∥MGp(t,s;ℝn):={E^[∫ts‍|η(r)|pdr]}1/p. Use the similar method in Lemma 43 of [27]; we can prove that υ∈MG2(t,s;ℝn) is a V-valued process; there exists {u=∑i=1N1Aiui}N∈ℕ; ui∈MG2,t(t,s;ℝn) is a V-valued process, Ai is a partition of ℬ(Ωt) such that u→v under probability measure P∈𝒫M. When υ(s)∈MG2,t(t,s;ℝn), we note that J(t,x;υ) is a deterministic function of (t,x) because b, hij, σj, Φ, f, and gij are deterministic functions, and B~s:=Bt+s-Bt is a G-Brownian motion. So we need to construct a sequence of admissible controls {υ~i(·)} of the form
(34)υ~si=∑j=1Niυsij1Aij
satisfying limi→∞J(t,x;υ~i(·))=u(t,x), where υij(·)∈MG2,t(t,s;ℝn) is a V-valued processes and {Aij}j=1Ni is a partition of ℬ(Ωt). Firstly, there exists {υk}k≥1⊂𝒰, such that u(t,x)=supk≥1J(t,x;υk). Then we define υ,υ′∈𝒰,
(35)(υ∨υ′)s={0,s∈[0,t];υs,s∈(t,T],on{J(t,x;υ)≥J(t,x;υ′)};υs′,s∈(t,T],on{J(t,x;υ)<J(t,x;υ′)}.
Therefore,
(36)J(t,x;υ∨υ′)≥J(t,x;υ)∨J(t,x;υ′).
Set υ¯1:=υ1∨υ1, υ¯k:=υ¯k-1∨υi, i≥2. So u(t,x)=limk→∞J(t,x;υ¯k). Without loss of generality, suppose E^[(u(t,x)-J(t,x;υ¯k))2]≤1/k,k≥1. We denote
(37)υ~sk=∑j,k=0Ni-1υ¯j,k(s)1Ajk,
where υ¯j,k∈MG2,t(t,s;ℝn) is a V-valued process and {Ajk}0≤j≤Nk-1 is a partition of ℬ(Ωt). Then we can suppose for k≥1, E^[∫tT‍|υ¯sk-υ~sk|2ds]≤1/Ck. From Proposition 22, we have
(38)E^[|J(t,x;υ¯k)-J(t,x;υ~k)|2]≤CE^[∫tT‍|υ¯sk-υ~sk|2ds]≤1k.
Therefore, E^[|u(t,x)-J(t,x;υ~k)|2]≤4/k. Then, we have
(39)J(t,x;υ~k)=∑j=0Nk-11AjkJ(t,x;υ¯j,k)≤u(t,x).
Now, we suppose that
(40)J(t,x,υ~)≤max0≤j≤Nk-1J(t,x;υ¯j,k)=J(t,x;υ¯j′,k).
Because E^[|J(t,x;υ~k)-u(t,x)|2]→0, we have
(41)u(t,x)=limk→∞J(t,x;υ¯j′,k),q.s..
Hence, E^[u(t,x)]=u(t,x). We have finished the proof.Lemma 24.
For anyt∈[0,T], x, x′∈ℝn, we have
(42)|u(t,x)-u(t,x′)|≤C|x-x′|,(43)|u(t,x)|≤C(1+|x|).Proof.
By Proposition22, we have for υ(·)∈𝒰,
(44)|J(t,x;υ(·))|≤C(1+|x|),|J(t,x;υ(·))-J(t,x′;υ(·))|≤C|x-x′|.
Then, ∀ε>0, there exist υ(·), υ′(·)∈𝒰 such that
(45)J(t,x;υ(·))≤u(t,x)≤J(t,x;υ(·))+ε,J(t,x′;υ′(·))≤u(t,x′)≤J(t,x′;υ′(·))+ε.
Now, we have
(46)-C(1+|x|)≤J(t,x;υ(·))≤u(t,x)≤J(t,x;υ(·))+ε≤C(1+|x|)+ε.
So, we get (43). Similarly, we obtain
(47)J(t,x;υ′(·))-J(t,x′;υ′(·))-ε≤u(t,x)-u(t,x′)≤J(t,x;υ(·))-J(t,x′;υ(·))+ε.
Then,
(48)-C|x-x′|-ε≤|u(t,x)-u(t,x′)|≤C|x-x′|+ε.
Thus, we have proved (42).Lemma 25.
For anyt∈[0,T], ζ∈LG2(Ωt;ℝn), and ζ is ℱtB measurable; we have ∀υ(·)∈𝒰,
(49)u(t,ζ)≥Ytt,ζ;υ.
Conversely, ∀ε>0, there exists a υ(·)∈𝒰, such that
(50)u(t,ζ)≤Ytt,ζ;υ+ε.Proof.
We already know thatu(t,x) is continuous with respect to x and Ytt,ζ;υ is continuous with respect to (ζ,υ(·)). We want to prove (49) and only need to discuss the simple random variables ζ of the form
(51)ζ=∑i=1N1Aixi,
and υ(·) of the form
(52)υ(·)=∑i=1N1Aiυi(·).
Here, i=1,2,…,N, xi∈ℝn, υi∈MG2,t(t,s;ℝn), and {Ai}i=1N is a ℬ(Ωt)-partition. Then, from the same technique used in the proof of Theorem 4.4 in [20], we have
(53)Ytt,ζ;υ=∑i=1N1AiYtt,xi;υi≤∑i=1N1Aiu(t,xi)=u(t,∑i=1N1Aixi)=u(t,ζ).
So we have proved (49). Now we prove (50) in a similar way. We first construct a random variable η∈LG2(Ωt;ℝn),
(54)η=∑i=1Nxsi1Ai,
where (Ai)i=1N is a ℬ(Ωt)-partition and xi∈ℝn, such that |η-ζ|≤ε/3C. Then, we have
(55)|Ytt,η;υ-Ytt,ζ;υ|≤ε3,|u(t,ζ)-u(t,η)|≤ε3,
for υ(·)∈𝒰. Now, we choose a control υi(·)∈MG2,t(t,s;ℝn), such that u(t,xi)≤Ytt,xi;υi+ε/3. Set υ(·):=∑i=1Nυi(·)1Ai. Finally, we get
(56)Yttζ;υ≥-|Ytt,η;υ-Ytt,ζ;υ|+Ytt,η;υ≥-ε3+∑i=1NYtt,xi;υi1Ai≥-ε3+∑i=1N(u(t,xi)-ε3)1Ai=-2ε3+∑i=1Nu(t,xi)1Ai=-2ε3+u(t,η)≥-ε+u(t,ζ).
So, we have (50).Now, we give a type of DPP for our stochastic optimal control problems. Firstly, we define a family of backward semigroups associated with the G-BSDE (29). Given the initial data (t,x), a positive number δ≤T-t, and a random variable η∈LGp(Ω;ℝ) with p>1, we set
(57)Gt,t+δt,x;υ[η]:=Yst,x;υ,
where (Yst,x;υ)t≤s≤t+δ is the solution of the following G-BSDE with the time horizon t+δ:
(58)Yst,x;υ=η+∫st+δ‍f(r,Xrt,x;υ,Yrt,x;υ,Zrt,x;υ,υr)dr-∫st+δ‍Zrt,x;υdBr-(KTt,x;υ-Ktt,x;υ)+∑i,j=1d∫st+δ‍gij(r,Xrt,x;υ,Yrt,x;υ,Zrt,x;υ,υr)d〈Bi,Bj〉r.
Obviously, for the solution Yt,x;υ of G-BSDE (29), we have
(59)Gt,Tt,x;υ[Φ(XTt,x;υ)]=Gt,t+δt,x;υ[Yt+δt,x;υ].
Then, we can obtain the DPP for our stochastic optimal control problems as follows.Theorem 26.
The value functionu(t,x) has the following proposition: for every 0≤δ≤T-t, we have
(60)u(t,x)=supυ(·)∈𝒰Gt,t+δt,x;υ[u(t+δ,Xt+δt,x;υ)].Proof.
We have(61)u(t,x)=supυ(·)∈𝒰Gt,Tt,x;υ[Φ(XTt,x;υ)]=supυ(·)∈𝒰Gt,t+δt,x;υ[Yt+δt+δ,Xt+δt,x;υ;υ].
Obviously, Xt+δt,x;υ is ℱt+δB measurable. So, by Lemma 25 and Theorem 19, we have
(62)u(t,x)≤supυ(·)∈𝒰Gt,t+δt,x;υ[u(t+δ,Xt+δt,x;υ)].
Besides, for ε>0, there exists an admissible control υ¯(·)∈𝒰 such that
(63)u(t+δ,Xt+δt,x;υ)≤Yt+δt+δ,Xt+δt,x;υ;υ¯+ε.
Then,
(64)u(t,x)≥supυ(·)∈𝒰Gt,t+δt,x;υ[u(t+δ,Xt+δt,x;υ)-ε]≥supυ(·)∈𝒰Gt,t+δt,x;υ[u(t+δ,Xt+δt,x;υ)]-Cε.
Because ε can be arbitrarily small, we get (60).Proposition 27.
u
(
t
,
x
) is 1/2-Hölder continuous in t.Proof.
For any given(t,x)∈[0,T]×ℝn and δ>0(t+δ≤T), from Theorem 26, we know that for ε>0, there exists a υ(·)∈𝒰 such that
(65)Gt,t+δt,x;υ[u(t+δ,Xt+δt,x;υ)]+ε≥u(t,x)≥Gt,t+δt,x;υ[u(t+δ,Xt+δt,x;υ)].
Then, we need to prove
(66)u(t,x)-u(t+δ,x)≤Cδ1/2(resp.,≥-Cδ1/2).
We only check the first inequality in (66). The second can be proved similarly. We have ∀ε>0,
(67)u(t,x)-u(t+δ,x)≤Iδ1+Iδ2+ε,
where
(68)Iδ1=Gt,t+δt,x;υ[u(t+δ,Xt+δt,x;υ)]-Gt,t+δt,x;υ[u(t+δ,x)],Iδ2=Gt,t+δt,x;υ[u(t+δ,x)]-u(t+δ,x).
From Proposition 21, we have
(69)E^t[|Xt+δt,x;υ-x|2]≤C(1+|x|2)δ.
By Proposition 22 and Lemma 24, we deduce that
(70)|Iδ1|≤[CE^t[|u(t+δ,Xt+δt,x;υ)-u(t+δ,x)|2]]1/2≤[CE^t[|Xt+δt,x;υ-x|2]]1/2≤C′δ1/2.
Based on the definition of Gt,t+δt,x;υ, we get
(71)Iδ2=E^t[u(t+δ,x)+∫tt+δ‍f(s,Xst,x;υ,Yst,x;υ,Zst,x;υ,υs)ds+∑i,j=1d∫tt+δ‍gij(s,Xst,x;υ,Yst,x;υ,Zst,x;υ,υs)d〈Bi,Bj〉s-∫tt+δ‍Zst,x;υdBs-(KTt,x;υ-Ktt,x;υ)]-u(t+δ,x)=E^t[∑i,j=1ds∫tt+δ‍f(s,Xst,x;υ,Yst,x;υ,Zst,x;υ,υs)ds+∑i,j=1d∫tt+δ‍gij(s,Xst,x;υ,Yst,x;υ,Zst,x;υ,υs)d〈Bi,Bj〉s]≤C′δ1/2×(1+E^t[∫tt+δ‍|Xst,x;υ|2+|Yst,x;υ|2+|Zst,x;υ|2ds]1/2).
By Proposition 22, we can prove the following inequality easily by the similar method in Proposition 3.5 of [11]
(72)E^t[∫tt+δ‍|Zst,x;υ|2ds]1/2≤C(1+|x|).
So we have Iδ2≤C′δ1/2. Hence, by (67) we have
(73)u(t,x)-u(t+δ,x)≤C′δ1/2+ε.
Let ε→0; we obtain the first inequality of (66). The proof is completed.


## 4. Value Function and Viscosity Solution of Fully Nonlinear Second-Order Partial Differential Equation

In this section, we consider the following fully nonlinear second-order partial differential equation(74)∂tu+F(Dx2u,Dxu,u,x,t)=0,(t,x)∈[0,T]×ℝn,u(T,x)=Φ(x),
where
(75)F(Dx2u,Dxu,u,x,t)=supυ∈V{G(H(Dx2u,Dxu,u,x,t,υ))+〈b(t,x,υ),Dxu〉+f(t,x,u,〈σ1(t,x),Dxu〉,…,〈σd(t,x),Dxu〉,υ)(H(Dx2u,Dxu,u,x,t,υ))},Hij(Dx2u,Dxu,u,x,t,υ)=〈Dx2u·σi(t,x,υ),σj(t,x,υ)〉+2〈Dxu,hij(t,x,υ)〉+2gij(t,x,u,〈σ1(t,x,υ),Dxu〉,…,〈σd(t,x,υ),Dxu〉,υ).Remark 28.
The definition and uniqueness of the viscosity solution of above second-order partial differential equation can be found in AppendixC in Peng [12]. So, we only need to prove that u(t,x) is a viscosity solution of (74). Besides, from the result of Section 3, we can have that u(t,x) is continuous in [0,T]×ℝn.Definition 29.
A real-valued continuous functionu(t,x)∈C([0,T]×ℝn), u(T,x)≤Φ(x), for any x∈ℝn, is called a viscosity subsolution (super-solution) of (74); if for all functions φ∈C2,3([0,T]×ℝn) satisfy φ≥u and φ(t,x)=u(t,x) at fixed (t,x)∈[0,T)×ℝn, we have
(76)∂tφ(t,x)+F(Dx2φ(t,x),Dxφ(t,x),φ(t,x),x,t)≥0(≤0).Theorem 30.
Under the assumptions (H3) and (H4), the value functionu(t,x) defined by (33) is a viscosity solution of (74).In order to prove the Theorem, we need three lemmas. Firstly, we set(77)F1(r,x,y,z,υ)=〈b(r,x,υ),Dxφ(r,x)〉+∂tφ(t,x)+f(r,x,y+φ(r,x),z+(〈σ1(t,x,υ),Dxφ(r,x)〉,…,〈σd(t,x,υ),Dxφ(r,x)〉),υ),F2ij(r,x,y,z,υ)=〈Dxφ(r,x),hij(r,x,υ)〉+12〈Dx2φ(r,x)σi(r,x,υ),σj(r,x,υ)〉+gij(r,x,y+φ(r,x),z+(〈σ1(t,x,υ),Dxφ(r,x)〉,…,〈σd(r,x,υ),Dxφ(r,x)〉),υ).
Then, we consider a G-BSDE defined on the interval [t,t+δ](0<δ≤T-t):
(78)Ys1,υ=∫st+δ‍F1(r,Xrt,x;υ,Yr1,υ,Zr1,υ,υr)dr+∫st+δ‍Zr1,υdBr-(Kt+δ1-Ks1)-∑i,j=1d∫st+δ‍F2ij(r,Xrt,x;υ,Yr1,υ,Zr1,υ,υr)d〈Bi,Bj〉r,
where υ(·)∈𝒰 and Xst,x;υ is defined by (24).Lemma 31.
Fors∈[t,t+δ], we have
(79)Gs,t+δt,x;υ[φ(Xt+δt,x;υ,t+δ)]-φ(Xst,x;υ,s)
which is the solution of (78).Proof.
From the definition ofGs,t+δt,x;υ, we know that Gs,t+δt,x;υ[φ(Xt+δt,x;υ,t+δ)] is the solution of G-BSDE (29) on [t,t+δ] with terminal condition φ(Xt+δt,x;υ,t+δ). Applying Itô’s formula to φ(Xst,x;υ,s), we can obtain the result.Now we construct a simple G-BSDE by replacing the driving processXst,x;υ by its deterministic initial value x as follows:
(80)Ys2,υ=∫st+δ‍F1(r,x,Yr2,υ,Zr2,υ,υr)dr+∑i,j=1d∫st+δ‍F2ij(r,x,Yr2,υ,Zr2,υ,υr)d〈Bi,Bj〉r-∫st+δ‍Zr2,υdBr-(Kt+δ2-Ks2).Lemma 32.
We have the following estimate, forυ(·)∈𝒰,
(81)|Yt1,υ-Yt2,υ|≤Cδ3/2,
where C is independent of the control processes υ(·).Proof.
By Proposition21, we have the estimate for p≥2(82)E^t[sups∈[t,t+δ]|Xst,x;υ-x|p]≤C(1+|x|p)δp/2.
By Proposition 18, we get for fixed p>2 and 2<p<β,
(83)|Yt1,υ-Yt2,υ|2≤E^[sups∈[t,t+δ]|Yt1,υ-Yt2,υ|2]≤C{E^[sups∈[t,t+δ]E^s[(∫tt+δ‍F^rdr)p]]2/p+E^[sups∈[t,t+δ]E^s[(∫tt+δ‍F^rdr)p]][sups∈[t,t+δ]E^s[(∫tt+δ‍F^rdr)p]]2/p},
where
(84)F^r=|F1(r,Xrt,x;υ,Yr2,υ,Zr2,υ,υr)-F1(r,x,Yr2,υ,Zr2,υ,υr)|+∑i,j=1d|F1i,j(r,Xrt,x;υ,Yr2,υ,Zr2,υ,υr)-F2i,j(r,x,Yr2,υ,Zr2,υ,υr)|.
It is easy to prove that
(85)F^r≤C|Xrt,x;υ-x|.
Then, we can deduce that |Yt1,υ-Yt2,υ|≤Cδ3/2.Lemma 33.
We have(86)supυ(·)∈𝒰Yt2,υ=Y0(t),
where Y0(·) is the solution of the following ODE:
(87)-dYs0=F0(s,x,Yr0,0)ds,s∈[t,t+δ],Yt+δ0=0,
where F0(r,x,y,z)=supυ∈V{F1(r,x,y,z,υ)+2G[(F2ij(r,x,y,z,υ))i,j=1d]}.Proof.
By Theorem16, we know that the G-BSDE (80) have a unique solution (Y,Z,K). Hence, there exists a process
(88)Vs2,υ=∑i,j=1d∫ts‍F2ij(r,x,Yr2,υ,Zr2,υ,υ)d〈Bi,Bj〉r-∫ts‍2G((F2ij(r,x,Yr2,υ,Zr2,υ,υ))i,j=1d)dr.
Here, Vs2,υ, s∈[t,t+δ] is a decreasing and continuous process by [28]. Besides, it satisfies E^[sups∈[t,t+δ]|Vs2,υ|β]<∞ obviously. So, Ys2,υ is the solution of the following G-BSDE:
(89)Ys2,υ=∫st+δ‍[12(F2ij(r,x,Yr2,υ,Zr2,υ,υr))i,j=1dF1(r,x,Yr2,υ,Zr2,υ,υr)+2G[(F2ij(r,x,Yr2,υ,Zr2,υ,υr))i,j=1d]12]dr-∫st+δ‍Zr2,υdBr-(Kt+δ2-Ks2)+Vt+δ2,υ-Vs2,υ,
where υ(·)∈𝒰. In addition, we have
(90)Yt0=∫st+δ‍F0(r,x,Yr0,Zr0)dr-∫st+δ‍Zr0dBr-(Kt+δ0-Ks0)+(Vt+δ0-Vs0),
where (Z,K,V)=0. By the comparison of Theorem 20 and the definition of F0, we have for υ(·)∈𝒰,
(91)Ys2,υ≤Ys0,s∈[t,t+δ].
On the other hand, there exists a measurable function υ′(r,x,y,z):[t,T]×ℝn×ℝ×ℝd×ℝ→V such that
(92)F0(r,x,y,z)=F1(r,x,y,z,υ′)+2G[(F2ij(r,x,y,0,υ′))i,j=1d].
Then, we have υ′(r,x,Yr0,Zr0)∈𝒰, and Yt0 is the solution of the following G-BSDE:
(93)Ys0=∫st+δ‍F1(r,x,Yr0,Zr0,υr′)dr+∑i,j=1d∫st+δ‍F2ij(r,x,Yr0,Zr0,υr′)d〈Bi,Bj〉r-∫st+δ‍Zr0dBr-(Kt+δ0-Ks0),
where Zr,υ0=0,
(94)Ks0=∑i,j=1d∫ts‍F2ij(r,x,Yr0,0,υ′)d〈Bi,Bj〉r-∫ts‍2G((F2ij(r,x,Yr0,0,υ′))i,j=1d)dr.
So, Yt0≤supυ(·)∈𝒰Yt2,υ. Now, we have proved the lemma.Then, we give the proof of Theorem30:Proof.
We setφ∈C2,3([0,T]×ℝn) and φ(t,x)=u(t,x) for fixed (t,x)∈[0,T]×ℝn. From Theorem 26, we know
(95)φ(t,x)=u(t,x)=supυ(·)∈𝒰Gt,t+δt,x;υ[u(Xt+δt,x,t+δ)].
By φ≥u(φ≤u) and the definition of G,
(96)supυ(·)∈𝒰{Gt,t+δt,x;υ[u(Xt+δt,x,t+δ)]-φ(t,x)}≥0(≤0).
Then, form Lemma 31,
(97)supυ(·)∈𝒰Yt1,υ≥0(≤0).
Besides, from Lemma 32,
(98)supυ(·)∈𝒰Yt2,υ≥Cδ3/2(≤Cδ3/2).
Finally, Lemma 33 implies
(99)Y0(t)≥Cδ3/2(≤Cδ3/2).
So, F0(r,x,0,0)≥0(≤0) and from the definition of viscosity solution of (74), we know u(t,x) is a viscosity solution of (74).



---

*Source: 564524-2013-09-08.xml*