# Representation Theorem for Generators of BSDEs Driven byG-Brownian Motion and Its Applications

Authors: Kun He; Mingshang Hu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We obtain a representation theorem for the generators of BSDEs driven byG-Brownian motions and then we use the representation theorem to get a converse comparison theorem for G-BSDEs and some equivalent results for nonlinear expectations generated by G-BSDEs.


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## Body

## 1. Introduction

Let(Ω,ℱ,P) be a probability space, and, for fixed T∈[0,+∞), let (Bt)0≤t≤T be a standard Brownian motion and let ℱt be the augmentation of σ{Bs,0≤s≤t}. Then Pardoux and Peng [1] introduced the backward stochastic differential equations (BSDEs) and proved the existence and uniqueness result of the BSDEs. In 1997, Peng [2] promoted g-expectations based on BSDEs. One of the important properties of g-expectations is comparison theorem or monotonicity. Chen [3] first considers a converse result of BSDEs under equal case. After that, Briand et al. [4] obtained a converse comparison theorem for BSDEs under general case. They also derived a representation theorem for the generator g. Following this paper, Jiang [5] discussed a more general representation theorem then, in his another paper [6], showed a more general converse comparison theorem. Here the representation theorem is an important method in solving the converse comparison problem and other problems (see Jiang [7]).Peng [8–13] defined the G-expectations and G-Brownian motions (G-BMs) and proved the representation theorem of G-expectation by a set of singular probabilities, which differs from nonlinear g-expectations because g-expectations are equivalent with a group of absolutely continuous probabilities with respect to the probability measure P. Soner et al. [14] obtained an existence and uniqueness result of 2 BSDEs. Recently, Hu et al. [15] proved another existence and uniqueness result on BSDEs driven by G-Brownian motions (G-BSDEs).An important advantage ofG-BSDEs is the easiness to define the nonlinear expectations. Hu et al. in [16] gave a comparison theorem for G-BSDEs and talked about the properties of corresponding nonlinear expectations. In this paper, we consider the representation theorem for generators of G-BSDEs and then consider the converse comparison theorem of G-BSDEs and some equivalent results for nonlinear expectations generated by G-BSDEs. In the following, in Section 2, we review some basic concepts and results about G-expectations. We give the representation theorem of G-BSDEs in Section 3. In Section 4, we consider the applications of representation theorem of G-BSDEs, which contain the converse comparison theorem and some equivalent results for nonlinear expectations generated by G-BSDEs.


## 2. Preliminaries

We review some basic notions and results ofG-expectation, the related spaces of random variables, and the backward stochastic differential equations driven by a G-Brownian motion. The readers may refer to [10, 13, 15, 17–19] for more details.Definition 1.
LetΩ be a given set and let ℋ be a vector lattice of real valued functions defined on Ω, namely, c∈ℋ for each constant c and |X|∈ℋ if X∈ℋ. ℋ is considered as the space of random variables. A sublinear expectation 𝔼^ on ℋ is a functional 𝔼^:ℋ→ℝ satisfying the following properties: for all X,Y∈ℋ, one has (a)
monotonicity: ifX≥Y, then 𝔼^[X]≥𝔼^[Y];(b)
constant preservation:𝔼^[c]=c;(c)
subadditivity:𝔼^[X+Y]≤𝔼^[X]+𝔼^[Y];(d)
positive homogeneity:𝔼^[λX]=λ𝔼^[X] for each λ≥0. (Ω,ℋ,𝔼^) is called a sublinear expectation space.Definition 2.
LetX1 and X2 be two n-dimensional random vectors defined, respectively, in sublinear expectation spaces (Ω1,ℋ1,𝔼^1) and (Ω2,ℋ2,𝔼^2). They are called identically distributed, denoted by X1=dX2, if 𝔼^1[φ(X1)]=𝔼^2[φ(X2)], for allφ∈Cb·Lip(ℝn), where Cb·Lip(ℝn) denotes the space of bounded and Lipschitz functions on ℝn.Definition 3.
In a sublinear expectation space(Ω,ℋ,𝔼^), a random vector Y=(Y1,…,Yn), Yi∈ℋ, is said to be independent of another random vector X=(X1,…,Xm), Xi∈ℋ under 𝔼^[·], denoted by Y⊥X, if for every test function φ∈Cb·Lip(ℝm×ℝn) one has 𝔼^[φ(X,Y)]=𝔼^[𝔼^[φ(x,Y)]x=X].Definition 4 (G-normal distribution).
Ad-dimensional random vector X=(X1,…,Xd) in a sublinear expectation space (Ω,ℋ,𝔼^) is called G-normally distributed if for each a,b≥0 one has
(1)aX+bX-=da2+b2X,
where X- is an independent copy of X; that is, X-=dX and X-⊥X. Here, the letter G denotes the function
(2)G(A):=12𝔼^[〈AX,X〉]:𝕊d⟶ℝ,
where 𝕊d denotes the collection of d×d symmetric matrices.Peng [13] showed that X=(X1,…,Xd) is G-normally distributed if and only if for each φ∈Cb·Lip(ℝd), u(t,x):=𝔼^[φ(x+tX)], (t,x)∈[0,∞)×ℝd, is the solution of the following G-heat equation:
(3)∂tu-G(Dx2u)=0,u(0,x)=φ(x).The functionG(·):𝕊d→ℝ is a monotonic, sublinear mapping on 𝕊d and G(A)=(1/2)𝔼^[〈AX,X〉]≤(1/2)|A|𝔼^[|X|2] implies that there exists a bounded, convex, and closed subset Γ⊂𝕊d+ such that
(4)G(A)=12supγ∈Γtr[γA],
where 𝕊d+ denotes the collection of nonnegative elements in 𝕊d.In this paper, we only consider nondegenerateG-normal distribution; that is, there exists some σ_2>0 such that G(A)-G(B)≥σ_2tr[A-B] for any A≥B.Definition 5.
(i) LetΩ=C0d(ℝ+) denote the space of ℝd-valued continuous functions on [0,∞) with ω0=0 and let Bt(ω)=ωt be the canonical process. Set
(5)Lip(Ω)≔{φ(Bt1,…,Btn):n≥1,t1,…,tn∈[0,∞),hhhφ∈Cb·Lip(ℝd×n)}.
Let G:𝕊d→ℝ be a given monotonic and sublinear function. G-expectation is a sublinear expectation defined by
(6)𝔼^[X]=𝔼~[φ(t1-t0ξ1,…,tm-tm-1ξm)],
for all X=φ(Bt1-Bt0,Bt2-Bt1,…,Btm-Btm-1), where ξ1,…,ξn 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 every i=1,…,m-1. The corresponding canonical process Bt=(Bti)i=1d is called a G-Brownian motion.
(ii) For each fixedt∈[0,∞), the conditional G-expectation 𝔼^t for ξ=φ(Bt1-Bt0,Bt2-Bt1,…,Btm-Btm-1)∈Lip(Ω), where without loss of generality we suppose ti=t, is defined by
(7)𝔼^t[φ(Bt1-Bt0,Bt2-Bt1,…,Btm-Btm-1)]=ψ(Bt1-Bt0,Bt2-Bt1,…,Bti-Bti-1),
where
(8)ψ(x1,…,xi)=𝔼^[φ(x1,…,xi,Bti+1-Bti,…,Btm-Btm-1)].For each fixedT>0, we set
(9)Lip(ΩT)≔{φ(Bt1,…,Btn):n≥1,t1,…,tn∈[0,T],hhhφ∈Cb·Lip(ℝd×n)}.
For each p≥1, we denote by LGp(Ω) (resp., LGp(ΩT)) the completion of Lip(Ω) (resp., Lip(ΩT)) under the norm ∥ξ∥p,G=(𝔼^[|ξ|p])1/p. It is easy to check that LGq(Ω)⊂LGp(Ω) for 1≤p≤q and 𝔼^t[·] can be extended continuously to LG1(Ω).For each fixeda∈ℝd, Bta=〈a,Bt〉 is a 1-dimensional Ga-Brownian motion, where Ga(α)=(1/2)(σaaT2α+-σ-aaT2α-), σaaT2=2G(aaT) and σ-aaT2=-2G(-aaT). Let πtN={t0N,…,tNN}, N=1,2,…, be a sequence of partitions of [0,t] such that μ(πtN)=max{|ti+1N-tiN|:i=0,…,N-1}→0; the quadratic variation process of Ba is defined by
(10)〈Ba〉t=limμ(πtN)→0∑j=0N-1‍(Btj+1Na-BtjNa)2.
For each fixed a,a-∈ℝd, the mutual variation process of Ba and Ba- is defined by
(11)〈Ba,Ba-〉t=14[〈Ba+a-〉t-〈Ba-a-〉t].Definition 6.
For fixedT>0, let MG0(0,T) be the collection of processes in the following form: for a given partition {t0,…,tN}=πT of [0,T],
(12)ηt(ω)=∑j=0N-1‍ξjI[tj,tj+1)(t),
where ξj∈Lip(Ωtj), j=0,1,2,…,N-1. For p≥1, one denotes by HGp(0,T), MGp(0,T) the completion of MG0(0,T) under the norms ∥η∥HGp={𝔼^[(∫0T‍|ηs|2ds)p/2]}1/p, ∥η∥MGp={𝔼^[∫0T‍|ηs|pds]}1/p, respectively.For eachη∈MG1(0,T), we can define the integrals ∫0T‍ηtdt and ∫0T‍ηtd〈Ba,Ba-〉t for each a, a-∈ℝd. For each η∈HGp(0,T;ℝd) with p≥1, we can define Itô's integral ∫0T‍ηtdBt.LetSG0(0,T)={h(t,Bt1∧t,…,Btn∧t):t1,…,tn∈[0,T],h∈Cb,Lip(ℝn+1)}. For p≥1 and η∈SG0(0,T), set ∥η∥SGp={𝔼^[supt∈[0,T]|ηt|p]}1/p. Denote by SGp(0,T) the completion of SG0(0,T) under the norm ∥·∥SGp.We consider the following type ofG-BSDEs (in this paper, we always use Einstein convention):
(13)Yt=ξ+∫tT‍f(s,Ys,Zs)ds+∫tT‍gij(s,Ys,Zs)d〈Bi,Bj〉s-∫tT‍ZsdBs-(KT-Kt),
where
(14)f(t,ω,y,z),gij(t,ω,y,z):[0,T]×ΩT×ℝ×ℝd⟶ℝ,
satisfy the following properties.(H1)
There exists someβ>1 such that for any y,z,f(·,·,y,z),gij(·,·,y,z)∈MGβ(0,T).(H2)
There exists someL>0 such that
(15)|f(t,y,z)-f(t,y′,z′)|+∑i,j=1d‍|gij(t,y,z)-gij(t,y′,z′)|≤L(|y-y′|+|z-z′|).For simplicity, we denote by𝔖Gα(0,T) the collection of processes (Y,Z,K) such that Y∈SGα(0,T), Z∈HGα(0,T;ℝd), K is a decreasing G-martingale with K0=0 and KT∈LGα(ΩT).Definition 7.
Letξ∈LGβ(ΩT) and f and gij satisfy (H1) and (H2) for some β>1. A triplet of processes (Y,Z,K) is called a solution of (13) if for some 1<α≤β the following properties hold:(a)
(Y,Z,K)∈𝔖Gα(0,T);(b)
Yt=ξ+∫tT‍f(s,Ys,Zs)ds+∫tT‍gij(s,Ys,Zs)d〈Bi,Bj〉s-∫tT‍ZsdBs-(KT-Kt).Theorem 8 (see [15]).
Assume thatξ∈LGβ(ΩT) and f and gij satisfy (H1) and (H2) for some β>1. Then, (13) has a unique solution (Y,Z,K). Moreover, for any 1<α<β, one has Y∈SGα(0,T), Z∈HGα(0,T;ℝd), and KT∈LGα(ΩT).We have the following estimates.Proposition 9 (see [15]).
Letξ∈LGβ(ΩT) and f, gij satisfy (H1) and (H2) for some β>1. Assume that (Y,Z,K)∈𝔖Gα(0,T) for some 1<α<β is a solution of (13). Then, there exists a constant Cα>0 depending on α, T, G, L such that
(16)|Yt|α≤Cα𝔼^t[|ξ|α+(∫tT|hs0|ds‍)α],𝔼^[(∫0T|Zs|2ds‍)α/2]≤Cα{𝔼^[supt∈[0,T]|Yt|α]+(𝔼^[supt∈[0,T]|Yt|α])1/2ggggggkkg×(𝔼^[(∫0Ths0ds‍)α])1/2},𝔼^[|Kt|α]≤Cα{𝔼^[supt∈[0,T]|Yt|α]+𝔼^[(∫0Ths0ds‍)α]},
where hs0=|f(s,0,0)|+∑i,j=1d‍|gij(s,0,0)|.Proposition 10 (see [15, 20]).
Letα≥1 and δ>0 be fixed. Then, there exists a constant C depending on α and δ such that
(17)𝔼^[supt∈[0,T]𝔼^t[|ξ|α]]≤C{(𝔼^[|ξ|α+δ])α/(α+δ)+𝔼^[|ξ|α+δ]},kkkkkkkkkkkkkkkkkk.kkk.∀ξ∈LGα+δ(ΩT).Theorem 11 (see [16]).
Let(Yl,Zl,Kl), l=1,2, be the solutions of the following G-BSDEs:
(18)Ytl=ξ+∫tTf(s,Ysl,Zsl)ds‍+∫tT‍gij(s,Ysl,Zsl)d〈Bi,Bj〉s+VTl-Vtl-∫tT‍ZsldBs-(KTl-Ktl),
where ξ∈LGβ(ΩT), f and gij satisfy (H1) and (H2) for some β>1 and (Vtl)t≤T are RCLL processes in MGβ(0,T) such that 𝔼^[supt∈[0,T]|Vtl|β]<∞. If   Vt1-Vt2 is an increasing process, then Yt1≥Yt2 for t∈[0,T].In this paper, we also need the following assumptions forG-BSDE (13).(H3)
For each fixed(ω,y,z)∈ΩT×ℝ×ℝd, t→f(t,ω,y,z) and t→gij(t,ω,y,z) are continuous.(H4)
For each fixed(t,y,z)∈[0,T)×ℝ×ℝd, f(t,y,z), gij(t,y,z)∈LGβ(Ωt), and
(19)limε→0+1ε𝔼^[∫tt+ε(|f(u,y,z)-f(t,y,z)|β∑i,j=1d‍‍jjjjjjjjjjjjjjjll;;;..jj+∑i,j=1d‍|gij(u,y,z)-gij(t,y,z)|β)du]=0.(H5)
For each(t,ω,y)∈[0,T]×ΩT×ℝ, f(t,ω,y,0)=gij(t,ω,y,0)=0.Assume thatξ∈LGβ(ΩT); f and gij satisfy (H1), (H2), and (H5) for some β>1. Let (YT,ξ,ZT,ξ,KT,ξ) be the solution of G-BSDE (13) corresponding to ξ, f, and gij on [0,T]. It is easy to check that YT,ξ=YT′,ξ on [0,T] for T′>T. Following [16], we can define consistent nonlinear expectation
(20)𝔼~t[ξ]=YtT,ξfort∈[0,T]
and set 𝔼~[ξ]=𝔼~0[ξ]=Y0T,ξ.


## 3. Representation Theorem of Generators ofG-BSDEs

We consider the following type ofG-FBSDEs:
(21)Xst,x=x+∫ts‍b(Xut,x)duXst,x=+∫tshij(Xut,x)d〈Bi,Bj〉u‍+∫tsσ(Xut,x)dBu‍,(22)εYst,x,y,p=y+〈p,Xt+εt,x-x〉εYst,x,y,p=+∫st+εf(u,Yut,x,y,pε,Zut,x,y,pε)du‍εYst,x,y,p=+∫st+ε‍gij(u,Yut,x,y,pε,Zut,x,y,pε)d〈Bi,Bj〉uεYst,x,y,p=-∫st+εZut,x,y,pεdBu‍-(Kt+εt,x,y,pε-Kst,x,y,pε),
where hij=hji and gij=gji, 1≤i,j≤d.We now give the main result in this section.Theorem 12.
Letb:ℝn→ℝn, hij:ℝn→ℝn, and σ:ℝn→ℝn×d be Lipschitz functions and let f and gij satisfy (H1), (H2), (H3), and (H4) for some β>1. Then, for each (t,x,y,p)∈[0,T)×ℝn×ℝ×ℝn and α∈(1,β), one has
(23)LGα-limε→0+1ε{Ytt,x,y,pε-y}=f(t,y,σT(x)p)+〈p,b(x)〉+2G((gij(t,y,σT(x)p)+〈p,hij(x)〉)i,j=1d).Proof.
For each fixed(t,x,y,p)∈[0,T)×ℝn×ℝ×ℝn, we write (Yε,Zε,Kε) instead of (Yt,x,y,pε,Zt,x,y,pε,Kt,x,y,pε) for simplicity. We have 𝔼^[|Xt+εt,x|γ]<∞ for each γ≥1 (see [16, 19]). Thus, by Theorem 8, G-BSDE (22) has a unique solution (Yε,Zε,Kε) and Ytε∈LGα(Ωt). We set, for s∈[t,t+ε],
(24)Y~sε=Ysε-(y+〈p,Xst,x-x〉),Z~sε=Zsε-σT(Xst,x)p,K~sε=Ksε.
Applying Itô’s formula to Y~sε on [t,t+ε], it is easy to verify that (Y~ε,Z~ε,K~ε) solves the following G-BSDE:
(25)Y~sε=∫st+ε‍f(u,Y~uε+y+〈p,Xut,x-x〉,hhhhhhhhZ~uε+σT(Xut,x)p)du+∫st+ε‍〈p,b(Xut,x)〉du+∫st+ε‍gij(u,Y~uε+y+〈p,Xut,x-x〉,hhhhhhhhhhlZ~uε+σT(Xut,x)p)d〈Bi,Bj〉u+∫st+ε‍〈p,hij(Xut,x)〉d〈Bi,Bj〉u-∫st+ε‍Z~uεdBu-(K~t+εε-K~sε).
From Proposition 9,
(26)|Y~sε|α≤Cα𝔼^s[(∫st+ε(|f(u,y+〈p,Xut,x-x〉,σT(Xut,x)p)|‍ffffffffjjjjjjfhhk+|〈p,b(Xut,x)〉|ffffffffjjjjjjhhfk+∑i,j=1d|gij(u,y+〈p,Xut,x-x〉,σT(Xut,x)p)|‍ffffffffjjjjjjfhh.∫+|〈p,hij(Xut,x)〉|)du)α],𝔼^[(∫tt+ε|Z~uε|2du‍)α/2]≤Cα{[sups∈[t,t+ε]|Y~sε|α]𝔼^[(∫tt+ε(|f(u,y+〈p,Xut,x-x〉,σT(Xut,x)p)|∑‍ffffffffjjjjjjfkkk+|〈p,b(Xut,x)〉|+|〈p,hij(Xut,x)〉|ffffffffjjjjjjfkkk+∑i,j=1d|gij(u,y+〈p,Xut,x-x〉,‍ffffffffjjjkkkkkkkkkkkkk∫∑σT(Xut,x)p)|)du)α]ffffffff+𝔼^[sups∈[t,t+ε]|Y~sε|α]},
hold for some constant Cα>0, only depending on α, T, G, and L. By Proposition 10 and the Lipschitz assumption, we obtain
(27)𝔼^[sups∈[t,t+ε]|Y~sε|α+(∫tt+ε|Z~uε|2du‍)α/2]≤C1εα𝔼^[1+(1ε∫tt+ε‍(|f(u,0,0)|β∑i,j=1d|gij(u,0,0)|β‍ffffffffffffgggggfkkk+∑i,j=1d|gij(u,0,0)|β‍)du)α/βffffffff.∑∑j∫tt+ε‍+sups∈[t,t+ε]|Xst,x|β],
where C1 is a constant depending on x, y, p, α, β, T, G, and L. Noting that 𝔼^[sups∈[t,t+ε]|Xst,x|β]≤C2(1+|x|β) (see [16, 19]), where C2 depends on T and L, and the following inequality holds:
(28)∫tt+ε‍(|f(u,0,0)|β+∑i,j=1d‍|gij(u,0,0)|β)du≤2β-1{ε(|f(t,0,0)|β+∑i,j=1d|gij(t,0,0)|β‍)fffffffffffffff+∫tt+ε(|f(u,0,0)-f(t,0,0)|β∑i,j=1d‍β‍fffffffffffffggggkkgff+∑i,j=1d‍|gij(u,0,0)-gij(t,0,0)|β)du}.
Together with assumption (H4), we get
(29)𝔼^[sups∈[t,t+ε]|Y~sε|α+(∫tt+ε‍|Z~uε|2du)α/2]≤C3εα,
where C3 depends on x, y, p, α, β, T, G, and L. Now, we prove (23). Let us consider
(30)1ε{Ytε-y}=1εY~tε=1ε𝔼^t[Y~tε+K~t+εε-K~tε]=1ε𝔼^t[∫tt+εf‍(u,y+〈p,Xut,x-x〉,σT(Xut,x)p)dufffff.k+∫tt+ε‍〈p,b(Xut,x)〉dufffff.k+∫tt+ε‍gij(u,y+〈p,Xut,x-x〉,σT(Xut,x)p)lllllllllllllllllll×d〈Bi,Bj〉ufffff.k+∫tt+ε‍〈p,hij(Xut,x)〉d〈Bi,Bj〉u]+Lε,
where
(31)Lε=1ε{𝔼^t[∫tt+εf(u,Y~uε+y+〈p,Xut,x-x〉,∫yyyyyyyyyyhhhhhhZ~uε+σT(Xut,x)p)dugggggggggg+∫tt+ε‍〈p,b(Xut,x)〉dugggggggggg+∫tt+εgij(u,Y~uε+y+〈p,Xut,x-x〉,‍yyyyyyyyyyhhjjjjjjjhhZ~uε+σT(Xut,x)p)d〈Bi,Bj〉ugggggggggg+∫tt+ε〈p,hij(Xut,x)〉d〈Bi,Bj〉u‍]Lε=1ε-𝔼^t[∫tt+ε‍f(u,y+〈p,Xut,x-x〉,σT(Xut,x)p)dugggggLε=1ε+∫tt+ε‍〈p,b(Xut,x)〉duggggLε=1εg+∫tt+ε‍gij(u,y+〈p,Xut,x-x〉,σT(Xut,x)p)hhhhhhhhhhhhhhhh×d〈Bi,Bj〉ugggggLε=1ε+∫tt+ε‍〈p,hij(Xut,x)〉d〈Bi,Bj〉u]}.
It is easy to check that |Lε|≤(C4/ε)𝔼^t[∫tt+ε‍(|Y~uε|+|Z~uε|)du], where C4 depends on G, L, and T. Thus, by (29), we get
(32)𝔼^[|Lε|α]≤C4αεα𝔼^[(∫tt+ε(|Y~uε|+|Z~uε|)du‍)α]≤2α-1C4αεα𝔼^[(∫tt+ε|Y~uε|du‍)α+(∫tt+ε|Z~uε|du‍)α]≤2α-1C4α{𝔼^[sups∈[t,t+ε]|Y~sε|α]+ε-α/2𝔼^[(∫tt+ε|Z~uε|2du‍)α/2]}≤2α-1C4αC3(εα+εα/2),
which implies LGα-limε→0+Lε=0. We set
(33)Mε=1ε{𝔼^t[∫tt+εf(u,y+〈p,Xut,x-x〉,σT(Xut,x)p)du‍ggggggggggk+∫tt+ε‍〈p,b(Xut,x)〉duggggggggggk+∫tt+ε‍gij(u,y+〈p,Xut,x-x〉,σT(Xut,x)p)gggggggggkkkkkkg×d〈Bi,Bj〉uggggggggggk+∫tt+ε〈p,hij(Xut,x)〉d〈Bi,Bj〉u‍]gggggggg-𝔼^t[∫tt+εf(u,y,σT(x)p)du‍+〈p,b(x)〉εgggggggfffffggg+∫tt+εgij(u,y,σT(x)p)‍d〈Bi,Bj〉ugggggggfffffggg+∫tt+ε〈p,hij(x)〉d〈Bi,Bj〉u‍]}.
By the Lipschitz condition, we can get |Mε|≤(C5/ε)𝔼^t[∫tt+ε‍|Xut,x-x|du], where C5 depends on p, G, L, and T. Noting that 𝔼^[sups∈[t,t+ε]|Xst,x-x|α]≤C6(1+|x|α)εα/2 (see [16, 19]), where C6 depends on L, G, and α, we obtain
(34)𝔼^[|Mε|α]≤C5α𝔼^[sups∈[t,t+ε]|Xst,x-x|α]≤C5αC6(1+|x|α)εα/2,
which implies LGα-limε→0+Mε=0. Now, we set
(35)Nε=1ε{𝔼^t[∫tt+εf(u,y,σT(x)p)du‍+〈p,b(x)〉εhhhhhhhhh.+∫tt+εgij(u,y,σT(x)p)d〈Bi,Bj〉u‍hhhhhhhhh.+∫tt+ε〈p,hij(x)〉d〈Bi,Bj〉u‍]hhhhhh.-𝔼^t[∫tt+εf(t,y,σT(x)p)du‍+〈p,b(x)〉εhhhhhhhhhhh+∫tt+εgij(t,y,σT(x)p)d〈Bi,Bj〉u‍hhhhhhhhhhh+∫tt+ε〈p,hij(x)〉d〈Bi,Bj〉u‍]}.
It is easy to deduce that |Nε|≤(C7/ε)𝔼^t[∫tt+ε‍(|f(u,y,σT(x)p)-f(t,y,σT(x)p)|+∑i,j=1d‍|gij(u,y,σT(x)p)-gij(t,y,σT(x)p)|)du], where C7 depends on G. Then,
(36)𝔼^[|Nε|α]≤C7α1ε𝔼^[∫tt+ε(|f(u,y,σT(x)p)-f(t,y,σT(x)p)|∑‍gggggggggggggggg+∑i,j=1d|gij(u,y,σT(x)p)‍hhhhhhhhhhhhhhhhhhh.∑-gij(t,y,σT(x)p)|)αdu]≤C7α(1ε𝔼^[∫tt+ε(|f(u,y,σT(x)p)∑‍∫gggggggggggggggg-f(t,y,σT(x)p)|gggggggggggggggkkg+∑i,j=1d|gij(u,y,σT(x)p)‍fffffffffffffffffffkkjjjk∑-gij(t,y,σT(x)p)|)βdu])α/β.
Take limit on both sides of the above inequality and use assumption (H4); then, we have
(37)LGα-limε→0+Nε=0.
On the other hand,
(38)𝔼^t[∫tt+εf(t,y,σT(x)p)du‍+〈p,b(x)〉εjjjjj+∫tt+εgij(t,y,σT(x)p)d〈Bi,Bj〉u‍jjjjj+∫tt+ε〈p,hij(x)〉d〈Bi,Bj〉u‍]=f(t,y,σT(x)p)ε+〈p,b(x)〉ε+𝔼^t[(gij(t,y,σT(x)p)+〈p,hij(x)〉)+𝔼^tkl×(〈Bi,Bj〉t+ε-〈Bi,Bj〉t)]=(f(t,y,σT(x)p)+〈p,b(x)〉∫k=+2G((gij(t,y,σT(x)p)+〈p,hij(x)〉)i,j=1d))ε.
Then, we have
(39)LGα-limε→0+1ε{Ytε-y}=f(t,y,σT(x)p)+〈p,b(x)〉+2G((gij(t,y,σT(x)p)+〈p,hij(x)〉)i,j=1d).
The proof is complete.


## 4. Some Applications

### 4.1. Converse Comparison Theorem forG-BSDEs

We consider the followingG-BSDEs:
(40)Ytl,ξ=ξ+∫tTfl(s,Ysl,ξ,Zsl,ξ)ds‍+∫tTgijl(s,Ysl,ξ,Zsl,ξ)d〈Bi,Bj〉s‍-∫tTZsl,ξdBs‍-(KTl,ξ-Ktl,ξ),l=1,2,
where gijl=gjil.We first generalized the comparison theorem in [16].Proposition 13.
Letfl and gijl satisfy (H1) and (H2) for some β>1, l=1,2. If f2-f1+2G((gij2-gij1)i,j=1d)≤0, then, for each ξ∈LGβ(ΩT), one has Yt1,ξ≥Yt2,ξ for t∈[0,T].Proof.
From the aboveG-BSDEs, we have
(41)Yt2,ξ=ξ+∫tTf2(s,Ys2,ξ,Zs2,ξ)ds‍+∫tTgij2(s,Ys2,ξ,Zs2,ξ)d〈Bi,Bj〉s‍-∫tTZs2,ξdBs‍-(KT2,ξ-Kt2,ξ)=ξ+∫tTf1(s,Ys2,ξ,Zs2,ξ)ds‍+∫tTgij1(s,Ys2,ξ,Zs2,ξ)d〈Bi,Bj〉s‍+VT-Vt-∫tTZs2,ξdBs‍-(KT2,ξ-Kt2,ξ),
where
(42)Vt=∫0t(f2-f1)(s,Ys2,ξ,Zs2,ξ)ds‍+∫0t(gij2-gij1)(s,Ys2,ξ,Zs2,ξ)d〈Bi,Bj〉s‍=∫0t(f2-f1+2G((gij2-gij1)i,j=1d))(s,Ys2,ξ,Zs2,ξ)ds‍+∫0t(gij2-gij1)(s,Ys2,ξ,Zs2,ξ)d〈Bi,Bj〉s‍-∫0t2G‍((gij2-gij1)i,j=1d)(s,Ys2,ξ,Zs2,ξ)ds.
By the assumption, it is easy to check that (Vt)t≤T is a decreasing process. Thus, using Theorem 11, we obtain Yt1,ξ≥Yt2,ξ for t∈[0,T].Remark 14.
Supposed=1 and let f1=10|z|, f2=|z|, g1=|z|, and g2=2|z|. It is easy to check that f2-f1+2G(g2-g1)≤0. Thus, f2-f1+2G((gij2-gij1)i,j=1d)≤0 does not imply f2≤f1 and (gij2)i,j=1d≤(gij1)i,j=1d.Now, we give the converse comparison theorem.Theorem 15.
Letfl and gijl satisfy (H1), (H2), (H3), (H4), and (H5) for some β>1, l=1,2. If Yt1,ξ≥Yt2,ξ for each t∈[0,T] and ξ∈LGβ(ΩT), then f2-f1+2G((gij2-gij1)i,j=1d)≤0 q.s..Proof.
For simplicity, we take the notation𝔼~tl[ξ]=Ytl,ξ, l=1,2. For each fixed (t,y,z)∈[0,T)×ℝ×ℝd, let us consider
(43)ηε=y+〈z,hij〉(〈Bi,Bj〉t+ε-〈Bi,Bj〉t)+〈z,Bt+ε-Bt〉,
where hij=hji∈ℝd. By Theorem 12, we have, for each α∈(1,β),
(44)LGα-limε→0+1ε(𝔼~tl[ηε]-y)=fl(t,y,z)+2G((gijl(t,y,z)+〈z,hij〉)i,j=1d).
Since 𝔼~t1[ηε]≥𝔼~t2[ηε],
(45)f1(t,y,z)+2G((gij1(t,y,z)+〈z,hij〉)i,j=1d)≥f2(t,y,z)+2G((gij2(t,y,z)+〈z,hij〉)i,j=1d)q.s.
Take a hij such that 〈z,hij〉=-gij1(t,y,z). Therefore, {f2-f1+2G((gij2-gij1)i,j=1d)}(t,y,z)≤0 q.s. By the assumptions (H2) and (H3), it is easy to deduce that f2-f1+2G((gij2-gij1)i,j=1d)≤0 q.s.In the following, we use the notation𝔼~tl[ξ]=Ytl,ξ, l=1,2.Corollary 16.
Letfl and gijl be deterministic functions and satisfy (H1), (H2), (H3), and (H5) for some β>1, l=1,2. If 𝔼~1[ξ]≥𝔼~2[ξ] for each ξ∈LGβ(ΩT), then f2-f1+2G((gij2-gij1)i,j=1d)≤0.Proof.
Takingηε as in Theorem 15, since fl and gijl are deterministic, we could get 𝔼~tl[ηε]=𝔼~l[ηε], for l=1, 2. And the proof in Theorem 15 still holds true.


### 4.2. Some Equivalent Relations

We consider the followingG-BSDE:
(46)Yt=ξ+∫tTf(s,Ys,Zs)ds‍+∫tTgij(s,Ys,Zs)d〈Bi,Bj〉s‍-∫tT‍ZsdBs-(KT-Kt),
where gij=gji. We use the notation 𝔼~t[ξ]=Yt.Proposition 17.
Letf and gij satisfy (H1), (H2), (H3), (H4), and (H5) for some β>1 and fix α∈(1,β). Then, one has(1)
𝔼~t[ξ+η]=𝔼~t[ξ]+ηfort∈[0,T],ξ∈LGα(ΩT),
andη∈LGα(Ωt)if and only if for eacht∈[0,T],y,y′∈ℝ,z∈ℝd,(47)f(t,y,z)-f(t,y′,z)+2G((gij(t,y,z)-gij(t,y′,z))i,j=1d)=0;(2)
𝔼~t[ξ+η]≤𝔼~t[ξ]+𝔼~t[η]fort∈[0,T],ξ∈LGα(ΩT),
andη∈LGα(ΩT)if and only if for eacht∈[0,T],y,y′∈ℝ,z,z′∈ℝd,(48)0≥f(t,y+y′,z+z′)-f(t,y,z)-f(t,y′,z′)+2G((gij(t,y+y′,z+z′)-gij(t,y,z)∑hhhhhhhhhh-gij(t,y′,z′))i,j=1d);(3)
𝔼~t[λξ+(1-λ)η]≤λ𝔼~t[ξ]+(1-λ)𝔼~t[η]fort∈[0,T],λ∈[0,1],ξ∈LGα(ΩT),
andη∈LGα(ΩT)if and only if for eacht∈[0,T],y,y′∈ℝ,z,z′∈ℝd,λ∈[0,1],(49)0≥f(t,λy+(1-λ)y′,λz+(1-λ)z′)-λf(t,y,z)-(1-λ)f(t,y′,z′)+2G((gij(t,λy+(1-λ)y′,λz+(1-λ)z′)∑gggggg-λgij(t,y,z)-(1-λ)gij(t,y′,z′))i,j=1d);(4)
𝔼~t[λξ]=λ𝔼~t[ξ]fort∈[0,T],λ≥0,
andξ∈LGα(ΩT)if and only if for eacht∈[0,T],y∈ℝ,z∈ℝd,λ≥0,(50)f(t,λy,λz)-λf(t,y,z)=2G((λgij(t,y,z)-gij(t,λy,λz))i,j=1d)=-2G((gij(t,λy,λz)-λgij(t,y,z))i,j=1d).Proof.
(1) “⇒” part. For each fixed t∈[0,T), y, y′∈ℝ, z∈ℝd, we take
(51)ξε=y+〈z,hij〉(〈Bi,Bj〉t+ε-〈Bi,Bj〉t)+〈z,Bt+ε-Bt〉,η=y′-y,
where hij=hji∈ℝd. Then, by Theorem 12 and 𝔼~t[ξε+η]=𝔼~t[ξε]+η, we can obtain
(52)f(t,y′,z)+2G((gij(t,y′,z)+〈z,hij〉)i,j=1d)=f(t,y,z)+2G((gij(t,y,z)+〈z,hij〉)i,j=1d).
We choose hij such that gij(t,y′,z)+〈z,hij〉=0, which implies (47).
“⇐” part. Let (Y,Z,K) be the solution of G-BSDE (46) corresponding to terminal condition ξ. We claim that (Ys+η,Zs,Ks)s∈[t,T] is the solution of G-BSDE (46) corresponding to terminal condition ξ+η on [t,T]. For this, we only need to check that, for s∈[t,T],
(53)∫sTf(u,Yu,Zu)du‍+∫sTgij(u,Yu,Zu)d〈Bi,Bj〉u‍=∫sTf(u,Yu+η,Zu)du‍+∫sTgij(u,Yu+η,Zu)d〈Bi,Bj〉u‍.
By (47) we can get
(54)∫sT(gij(u,Yu,Zu)-gij(u,Yu+η,Zu))d〈Bi,Bj〉u‍-2∫sTG((gij(u,Yu,Zu)-gij(u,Yu+η,Zu))i,j=1d)du‍=∫sT(gij(u,Yu,Zu)-gij(u,Yu+η,Zu))d〈Bi,Bj〉u‍+∫sT(f(u,Yu,Zu)-f(u,Yu+η,Zu))du‍≤0,∫sT(gij(u,Yu+η,Zu)-gij(u,Yu,Zu))d〈Bi,Bj〉u‍-2∫sTG((gij(u,Yu+η,Zu)-gij(u,Yu,Zu))i,j=1d)du‍=∫sT(gij(u,Yu+η,Zu)-gij(u,Yu,Zu))d〈Bi,Bj〉u‍+∫sT(f(u,Yu+η,Zu)-f(u,Yu,Zu))du‍≤0,
which implies (53). The proof of (1) is complete.
(2) “⇒” part. For each fixed t∈[0,T), y, y′∈ℝ, z, z′∈ℝd, we consider ξε=y+〈z,hij〉(〈Bi,Bj〉t+ε-〈Bi,Bj〉t)+〈z,Bt+ε-Bt〉 and ηε=y′+〈z′,hij′〉(〈Bi,Bj〉t+ε-〈Bi,Bj〉t)+〈z′,Bt+ε-Bt〉, where hij=hji∈ℝd and hij′=hji′∈ℝd. Then, by Theorem 12 and 𝔼~t[ξε+ηε]=𝔼~t[ξε]+𝔼~t[ηε], we obtain
(55)f(t,y+y′,z+z′)+2G((gij(t,y+y′,z+z′)∑hhhhhhhh+〈z,hij〉+〈z′,hij′〉)i,j=1d)≤f(t,y,z)+f(t,y′,z′)+2G((gij(t,y,z)+〈z,hij〉)i,j=1d)+2G((gij(t,y′,z′)+〈z′,hij′〉)i,j=1d).
We choose hij, hij′ such that gij(t,y,z)+〈z,hij〉=0 and gij(t,y′,z′)+〈z′,hij′〉=0, which implies (48).
“⇐” part. Let (Y,Z,K) and (Y′,Z′,K′) be the solutions of G-BSDE (46) corresponding to terminal condition ξ and η, respectively. Then, (Y+Y′,Z+Z′,K) solves the following G-BSDE:
(56)Yt+Yt′=ξ+η+∫tTf(s,Ys+Ys′,Zs+Zs′)ds‍+∫tTgij(s,Ys+Ys′,Zs+Zs′)d〈Bi,Bj〉s‍+VT-Vt-∫tT(Zs+Zs′)dBs‍-(KT-Kt),
where
(57)Vt=-Kt′-∫0t(f(s,Ys+Ys′,Zs+Zs′)‍vvvvvvvvvv-f(s,Ys,Zs)-f(s,Ys′,Zs′))ds-∫0t(gij(s,Ys+Ys′,Zs+Zs′)-gij(s,Ys,Zs)‍hhhhhh-gij(s,Ys′,Zs′))d〈Bi,Bj〉s=-Kt′-{∫0t(gij(s,Ys+Ys′,Zs+Zs′)‍∑kkkkkkkkkkkk.-gij(s,Ys,Zs)-gij(s,Ys′,Zs′))d〈Bi,Bj〉skkkkkkkk-2∫0t‍G((gij(s,Ys+Ys′,Zs+Zs′)∑kkkkkkkkkkkkkkkkk.kll-gij(s,Ys,Zs)kkkkkkkkkkkkkkkkkkll∑-gij(s,Ys′,Zs′))i,j=1d)ds}-∫0t{f(s,Ys+Ys′,Zs+Zs′)∑‍hhhhhhh-f(s,Ys,Zs)-f(s,Ys′,Zs′)kkkkkkkl+2G((gij(s,Ys+Ys′,Zs+Zs′)-gij(s,Ys,Zs)∑kkkkkkkkkkkkkkk-gij(s,Ys′,Zs′))i,j=1d)}ds.
By (48), it is easy to check that Vt is an increasing process. Then, by Theorem 11, we can get 𝔼~t[ξ+η]≤𝔼~t[ξ]+𝔼~t[η]. The proof of (2) is complete.
Finally, we could prove (3) as in (2) and (4) as in (1).Proposition 18.
One has the following.
(1)
If
G
(
A
)
+
G
(
-
A
)
>
0
for any
A
∈
𝕊
d
and
A
≠
0
, then (47) holds if and only if
f
and
g
i
j
are independent of
y
.
(2)
If there exists an
A
∈
𝕊
d
with
A
≠
0
such that
G
(
A
)
+
G
(
-
A
)
=
0
and
G
(
A
)
≠
0
, then, for any fixed
g
(
t
,
y
,
z
)
satisfying (H1)–(H5), one has
f
(
t
,
y
,
z
)
=
-
2
G
(
A
)
g
(
t
,
y
,
z
)
and
(
g
i
j
(
t
,
y
,
z
)
)
i
,
j
=
1
d
=
g
(
t
,
y
,
z
)
A
satisfying (47).Proof.
It is easy to verify (2), and we only need to prove (1). If (47) holds, it is easy to check that G((gij(t,y,z)-gij(t,0,z))i,j=1d)+G((gij(t,0,z)-gij(t,y,z))i,j=1d)=0 holds. Then, from the assumption, we get gij(t,y,z)=gij(t,0,z). Therefore, by (47), we have f(t,y,z)=f(t,0,z), which implies that f and gij are independent of y. The converse part is obvious.


## 4.1. Converse Comparison Theorem forG-BSDEs

We consider the followingG-BSDEs:
(40)Ytl,ξ=ξ+∫tTfl(s,Ysl,ξ,Zsl,ξ)ds‍+∫tTgijl(s,Ysl,ξ,Zsl,ξ)d〈Bi,Bj〉s‍-∫tTZsl,ξdBs‍-(KTl,ξ-Ktl,ξ),l=1,2,
where gijl=gjil.We first generalized the comparison theorem in [16].Proposition 13.
Letfl and gijl satisfy (H1) and (H2) for some β>1, l=1,2. If f2-f1+2G((gij2-gij1)i,j=1d)≤0, then, for each ξ∈LGβ(ΩT), one has Yt1,ξ≥Yt2,ξ for t∈[0,T].Proof.
From the aboveG-BSDEs, we have
(41)Yt2,ξ=ξ+∫tTf2(s,Ys2,ξ,Zs2,ξ)ds‍+∫tTgij2(s,Ys2,ξ,Zs2,ξ)d〈Bi,Bj〉s‍-∫tTZs2,ξdBs‍-(KT2,ξ-Kt2,ξ)=ξ+∫tTf1(s,Ys2,ξ,Zs2,ξ)ds‍+∫tTgij1(s,Ys2,ξ,Zs2,ξ)d〈Bi,Bj〉s‍+VT-Vt-∫tTZs2,ξdBs‍-(KT2,ξ-Kt2,ξ),
where
(42)Vt=∫0t(f2-f1)(s,Ys2,ξ,Zs2,ξ)ds‍+∫0t(gij2-gij1)(s,Ys2,ξ,Zs2,ξ)d〈Bi,Bj〉s‍=∫0t(f2-f1+2G((gij2-gij1)i,j=1d))(s,Ys2,ξ,Zs2,ξ)ds‍+∫0t(gij2-gij1)(s,Ys2,ξ,Zs2,ξ)d〈Bi,Bj〉s‍-∫0t2G‍((gij2-gij1)i,j=1d)(s,Ys2,ξ,Zs2,ξ)ds.
By the assumption, it is easy to check that (Vt)t≤T is a decreasing process. Thus, using Theorem 11, we obtain Yt1,ξ≥Yt2,ξ for t∈[0,T].Remark 14.
Supposed=1 and let f1=10|z|, f2=|z|, g1=|z|, and g2=2|z|. It is easy to check that f2-f1+2G(g2-g1)≤0. Thus, f2-f1+2G((gij2-gij1)i,j=1d)≤0 does not imply f2≤f1 and (gij2)i,j=1d≤(gij1)i,j=1d.Now, we give the converse comparison theorem.Theorem 15.
Letfl and gijl satisfy (H1), (H2), (H3), (H4), and (H5) for some β>1, l=1,2. If Yt1,ξ≥Yt2,ξ for each t∈[0,T] and ξ∈LGβ(ΩT), then f2-f1+2G((gij2-gij1)i,j=1d)≤0 q.s..Proof.
For simplicity, we take the notation𝔼~tl[ξ]=Ytl,ξ, l=1,2. For each fixed (t,y,z)∈[0,T)×ℝ×ℝd, let us consider
(43)ηε=y+〈z,hij〉(〈Bi,Bj〉t+ε-〈Bi,Bj〉t)+〈z,Bt+ε-Bt〉,
where hij=hji∈ℝd. By Theorem 12, we have, for each α∈(1,β),
(44)LGα-limε→0+1ε(𝔼~tl[ηε]-y)=fl(t,y,z)+2G((gijl(t,y,z)+〈z,hij〉)i,j=1d).
Since 𝔼~t1[ηε]≥𝔼~t2[ηε],
(45)f1(t,y,z)+2G((gij1(t,y,z)+〈z,hij〉)i,j=1d)≥f2(t,y,z)+2G((gij2(t,y,z)+〈z,hij〉)i,j=1d)q.s.
Take a hij such that 〈z,hij〉=-gij1(t,y,z). Therefore, {f2-f1+2G((gij2-gij1)i,j=1d)}(t,y,z)≤0 q.s. By the assumptions (H2) and (H3), it is easy to deduce that f2-f1+2G((gij2-gij1)i,j=1d)≤0 q.s.In the following, we use the notation𝔼~tl[ξ]=Ytl,ξ, l=1,2.Corollary 16.
Letfl and gijl be deterministic functions and satisfy (H1), (H2), (H3), and (H5) for some β>1, l=1,2. If 𝔼~1[ξ]≥𝔼~2[ξ] for each ξ∈LGβ(ΩT), then f2-f1+2G((gij2-gij1)i,j=1d)≤0.Proof.
Takingηε as in Theorem 15, since fl and gijl are deterministic, we could get 𝔼~tl[ηε]=𝔼~l[ηε], for l=1, 2. And the proof in Theorem 15 still holds true.


## 4.2. Some Equivalent Relations

We consider the followingG-BSDE:
(46)Yt=ξ+∫tTf(s,Ys,Zs)ds‍+∫tTgij(s,Ys,Zs)d〈Bi,Bj〉s‍-∫tT‍ZsdBs-(KT-Kt),
where gij=gji. We use the notation 𝔼~t[ξ]=Yt.Proposition 17.
Letf and gij satisfy (H1), (H2), (H3), (H4), and (H5) for some β>1 and fix α∈(1,β). Then, one has(1)
𝔼~t[ξ+η]=𝔼~t[ξ]+ηfort∈[0,T],ξ∈LGα(ΩT),
andη∈LGα(Ωt)if and only if for eacht∈[0,T],y,y′∈ℝ,z∈ℝd,(47)f(t,y,z)-f(t,y′,z)+2G((gij(t,y,z)-gij(t,y′,z))i,j=1d)=0;(2)
𝔼~t[ξ+η]≤𝔼~t[ξ]+𝔼~t[η]fort∈[0,T],ξ∈LGα(ΩT),
andη∈LGα(ΩT)if and only if for eacht∈[0,T],y,y′∈ℝ,z,z′∈ℝd,(48)0≥f(t,y+y′,z+z′)-f(t,y,z)-f(t,y′,z′)+2G((gij(t,y+y′,z+z′)-gij(t,y,z)∑hhhhhhhhhh-gij(t,y′,z′))i,j=1d);(3)
𝔼~t[λξ+(1-λ)η]≤λ𝔼~t[ξ]+(1-λ)𝔼~t[η]fort∈[0,T],λ∈[0,1],ξ∈LGα(ΩT),
andη∈LGα(ΩT)if and only if for eacht∈[0,T],y,y′∈ℝ,z,z′∈ℝd,λ∈[0,1],(49)0≥f(t,λy+(1-λ)y′,λz+(1-λ)z′)-λf(t,y,z)-(1-λ)f(t,y′,z′)+2G((gij(t,λy+(1-λ)y′,λz+(1-λ)z′)∑gggggg-λgij(t,y,z)-(1-λ)gij(t,y′,z′))i,j=1d);(4)
𝔼~t[λξ]=λ𝔼~t[ξ]fort∈[0,T],λ≥0,
andξ∈LGα(ΩT)if and only if for eacht∈[0,T],y∈ℝ,z∈ℝd,λ≥0,(50)f(t,λy,λz)-λf(t,y,z)=2G((λgij(t,y,z)-gij(t,λy,λz))i,j=1d)=-2G((gij(t,λy,λz)-λgij(t,y,z))i,j=1d).Proof.
(1) “⇒” part. For each fixed t∈[0,T), y, y′∈ℝ, z∈ℝd, we take
(51)ξε=y+〈z,hij〉(〈Bi,Bj〉t+ε-〈Bi,Bj〉t)+〈z,Bt+ε-Bt〉,η=y′-y,
where hij=hji∈ℝd. Then, by Theorem 12 and 𝔼~t[ξε+η]=𝔼~t[ξε]+η, we can obtain
(52)f(t,y′,z)+2G((gij(t,y′,z)+〈z,hij〉)i,j=1d)=f(t,y,z)+2G((gij(t,y,z)+〈z,hij〉)i,j=1d).
We choose hij such that gij(t,y′,z)+〈z,hij〉=0, which implies (47).
“⇐” part. Let (Y,Z,K) be the solution of G-BSDE (46) corresponding to terminal condition ξ. We claim that (Ys+η,Zs,Ks)s∈[t,T] is the solution of G-BSDE (46) corresponding to terminal condition ξ+η on [t,T]. For this, we only need to check that, for s∈[t,T],
(53)∫sTf(u,Yu,Zu)du‍+∫sTgij(u,Yu,Zu)d〈Bi,Bj〉u‍=∫sTf(u,Yu+η,Zu)du‍+∫sTgij(u,Yu+η,Zu)d〈Bi,Bj〉u‍.
By (47) we can get
(54)∫sT(gij(u,Yu,Zu)-gij(u,Yu+η,Zu))d〈Bi,Bj〉u‍-2∫sTG((gij(u,Yu,Zu)-gij(u,Yu+η,Zu))i,j=1d)du‍=∫sT(gij(u,Yu,Zu)-gij(u,Yu+η,Zu))d〈Bi,Bj〉u‍+∫sT(f(u,Yu,Zu)-f(u,Yu+η,Zu))du‍≤0,∫sT(gij(u,Yu+η,Zu)-gij(u,Yu,Zu))d〈Bi,Bj〉u‍-2∫sTG((gij(u,Yu+η,Zu)-gij(u,Yu,Zu))i,j=1d)du‍=∫sT(gij(u,Yu+η,Zu)-gij(u,Yu,Zu))d〈Bi,Bj〉u‍+∫sT(f(u,Yu+η,Zu)-f(u,Yu,Zu))du‍≤0,
which implies (53). The proof of (1) is complete.
(2) “⇒” part. For each fixed t∈[0,T), y, y′∈ℝ, z, z′∈ℝd, we consider ξε=y+〈z,hij〉(〈Bi,Bj〉t+ε-〈Bi,Bj〉t)+〈z,Bt+ε-Bt〉 and ηε=y′+〈z′,hij′〉(〈Bi,Bj〉t+ε-〈Bi,Bj〉t)+〈z′,Bt+ε-Bt〉, where hij=hji∈ℝd and hij′=hji′∈ℝd. Then, by Theorem 12 and 𝔼~t[ξε+ηε]=𝔼~t[ξε]+𝔼~t[ηε], we obtain
(55)f(t,y+y′,z+z′)+2G((gij(t,y+y′,z+z′)∑hhhhhhhh+〈z,hij〉+〈z′,hij′〉)i,j=1d)≤f(t,y,z)+f(t,y′,z′)+2G((gij(t,y,z)+〈z,hij〉)i,j=1d)+2G((gij(t,y′,z′)+〈z′,hij′〉)i,j=1d).
We choose hij, hij′ such that gij(t,y,z)+〈z,hij〉=0 and gij(t,y′,z′)+〈z′,hij′〉=0, which implies (48).
“⇐” part. Let (Y,Z,K) and (Y′,Z′,K′) be the solutions of G-BSDE (46) corresponding to terminal condition ξ and η, respectively. Then, (Y+Y′,Z+Z′,K) solves the following G-BSDE:
(56)Yt+Yt′=ξ+η+∫tTf(s,Ys+Ys′,Zs+Zs′)ds‍+∫tTgij(s,Ys+Ys′,Zs+Zs′)d〈Bi,Bj〉s‍+VT-Vt-∫tT(Zs+Zs′)dBs‍-(KT-Kt),
where
(57)Vt=-Kt′-∫0t(f(s,Ys+Ys′,Zs+Zs′)‍vvvvvvvvvv-f(s,Ys,Zs)-f(s,Ys′,Zs′))ds-∫0t(gij(s,Ys+Ys′,Zs+Zs′)-gij(s,Ys,Zs)‍hhhhhh-gij(s,Ys′,Zs′))d〈Bi,Bj〉s=-Kt′-{∫0t(gij(s,Ys+Ys′,Zs+Zs′)‍∑kkkkkkkkkkkk.-gij(s,Ys,Zs)-gij(s,Ys′,Zs′))d〈Bi,Bj〉skkkkkkkk-2∫0t‍G((gij(s,Ys+Ys′,Zs+Zs′)∑kkkkkkkkkkkkkkkkk.kll-gij(s,Ys,Zs)kkkkkkkkkkkkkkkkkkll∑-gij(s,Ys′,Zs′))i,j=1d)ds}-∫0t{f(s,Ys+Ys′,Zs+Zs′)∑‍hhhhhhh-f(s,Ys,Zs)-f(s,Ys′,Zs′)kkkkkkkl+2G((gij(s,Ys+Ys′,Zs+Zs′)-gij(s,Ys,Zs)∑kkkkkkkkkkkkkkk-gij(s,Ys′,Zs′))i,j=1d)}ds.
By (48), it is easy to check that Vt is an increasing process. Then, by Theorem 11, we can get 𝔼~t[ξ+η]≤𝔼~t[ξ]+𝔼~t[η]. The proof of (2) is complete.
Finally, we could prove (3) as in (2) and (4) as in (1).Proposition 18.
One has the following.
(1)
If
G
(
A
)
+
G
(
-
A
)
>
0
for any
A
∈
𝕊
d
and
A
≠
0
, then (47) holds if and only if
f
and
g
i
j
are independent of
y
.
(2)
If there exists an
A
∈
𝕊
d
with
A
≠
0
such that
G
(
A
)
+
G
(
-
A
)
=
0
and
G
(
A
)
≠
0
, then, for any fixed
g
(
t
,
y
,
z
)
satisfying (H1)–(H5), one has
f
(
t
,
y
,
z
)
=
-
2
G
(
A
)
g
(
t
,
y
,
z
)
and
(
g
i
j
(
t
,
y
,
z
)
)
i
,
j
=
1
d
=
g
(
t
,
y
,
z
)
A
satisfying (47).Proof.
It is easy to verify (2), and we only need to prove (1). If (47) holds, it is easy to check that G((gij(t,y,z)-gij(t,0,z))i,j=1d)+G((gij(t,0,z)-gij(t,y,z))i,j=1d)=0 holds. Then, from the assumption, we get gij(t,y,z)=gij(t,0,z). Therefore, by (47), we have f(t,y,z)=f(t,0,z), which implies that f and gij are independent of y. The converse part is obvious.



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*Source: 342038-2013-12-26.xml*