getdoc7888. 231KB Jun 04 2011 12:04:32 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 15 (2010), Paper no. 67, pages 2041–2068.

Journal URL

http://www.math.washington.edu/~ejpecp/

Martingale Property and Capacity under G-Framework

Jing Xu

∗

School of Economics and Business Administration Chongqing University Chongqing 400030

Email:Xujing@yahoo.com.cn

Bo Zhang

†

Center for Applied Statistics and School of Statistics Renmin University of China, Beijing, 100872, P.R.China

Email:mabzhang@gmail.co

Abstract

The main purpose of this article is to study the symmetric martingale property and capacity defined by G-expectation introduced by Peng (cf. http://arxiv.org/PS_cache/math/pdf/0601/

0601035v2.pdf) in 2006. We show that the G-capacity can not be dynamic, and also demonstrate the relationship between symmetric G-martingale and the martingale under linear expectation. Based on these results and path-wise analysis, we obtain the martingale characterization the-orem for G Brownian motion without Markovian assumption. This thethe-orem covers the Lèvy’s martingale characterization theorem for Brownian motion, and it also gives a different method to prove Lèvy’s theorem.

Key words:G-Brownian motion, G-expectation, Martingale characterization, Capacity. AMS 2000 Subject Classification:Primary 60H05, 60H10.

Submitted to EJP on January 3, 2010, final version accepted November 9, 2010.

∗_{Supported by National Natural Science Foundation of China (Grant No. 10901168), Natural Science Foundation} Project of CQ CSTC (Grant No. 2009BB2039) and the Project of Chinese Ministry of Education (Grant No. 09YJCZH122).

†_{Corresponding Author. Supported by NNSFC(Grant No. 10771214 and 71071155), MOE Humanities Social Science}

Key Research Institute in University Foundation (Grant No. 07JJD910244), Fundamental Research Funds for the Central Universities, and Research Funds of Renmin University of China (Grant No.10XNL007)

1

I

NTRODUCTION

In 1969, Robert C. Merton introduced stochastic calculus to finance, see [20], and indeed to the

broader field of economics, beginning an amazing decade of developments. The most famous pricing

formula for the European call option was given by Black and Scholes in 1973, see[2]. As these

developments unfolded, Cox and Ross [6] notice that, without loss of generality, to price some

derivative security as an option, one would get the correct result by assuming that all of the securities have the same expected rate of return. This is the “risk neutral” pricing method. This notion is first

given by Harrison and Kreps[14], who formalize (under conditions) the near equivalence of the

absence arbitrage with the existence of some new “risk neutral” probability measure under which all expected rates of return are indeed equal to the current risk free rate. To get this “risk-neutral" probability measure, Harrison and Kreps apply Girsanov’s theorem to change the measure.

Black & Scholes’s result and Cox’s result have been widely used since it appeared. But their work deeply depends on certain assumptions, e.g., the interest rate and the volatility of the stock price remain constant and known. In fact, the interest rate and the volatility of the stock price are not always constant and known, which are called mean uncertainty and volatility uncertainty. As for the mean uncertainty, Girsanov’s theorem or Peng’s g-expectation is a powerful tool to solve the

problem, see[3]and [11]. How to deal with the volatility uncertainty is a big problem, see[1],

[15], [19], [27]. As in [9], the main difficulty is that we have to deal with a series of probability

measures which are not absolutely continuous with respect to one single probability measure. It shows that this problem can not be solved in a given probability space.

In 2006, Peng made a change to the heat equation that Brownian motion satisfies, see[23],[24],

and constructed the G-normal distribution via the modified heat equation. With this G-normal dis-tribution, a nonlinear expectation is given which is called G-expectation and the related conditional expectation is constructed, which is a kind of dynamic coherent risk measure introduced by Delbaen

in[7]. Under this framework, the canonical process is a G-Brownian motion. The stochastic calculus

of Itô’s type with respect to the G-Brownian motion and the related Itô’s formula are also derived. G-Brownian motion, different from Brownian motion in the classical case, is not defined on a given probability space.

It is interesting to get a pricing formula (Black-Scholes formula) in G-framework. To do this, it is important to give the Girsanov theorem in this framework. Its proof depends heavily on the martingale characterization of Brownian motion, due to Lèvy. This theorem enables us to recognize a Brownian motion just with one or two martingale properties of a process.

In order to get the Girsanov theorem, we need to describe this G-Brownian motion by its martingale properties. That is the martingale characterization theorem for G-Brownian motion.

In [29], the authors give the martingale characterization theorem for G-Brownian motion under

Markovian condition, but this condition is not so convenient in applications.

In this paper, we define capacity via G-expectation, and state that G-expectation is not filtration consistent. Then we investigate the relationship between symmetric G-martingales and martingales under linear expectation, when the corresponding G-heat equation is uniformly parabolic. Based on these results, we give the martingale characterization theorem for G-Brownian motion without

Markovian condition which improves the related result in [29]. The current result extends the

classical Lèvy’s theorem. Additionally, we give a different method to prove Lèvy’s theorem.

give the martingale characterization of G-Brownian motion when the corresponding G-heat equation is uniformly parabolic, see in section 5. G-expectation theory has received more attention since

Peng’s basic paper appeared, see[23],[24],[25]. Soner et al[21]study the G-martingale problem

under some condition, and investigate the representation theorem for all G-martingales including the non-symmetric martingale based on a class of backward stochastic differential equations. In the current work, we are focused on symmetric G-martingale, our method is totally different from

the method in[21]. As for the martingale characterization for G-Brownian motion, our method is

different from [29], which is based on viscosity solution theory for nonlinear parabolic equation.

But in this paper, the path wise analysis is important to underly our result, which is a different

approach from [29]. Meanwhile, the result in this paper is very useful for further application in

finance, especially for option pricing problems with volatility uncertainty. By using the result in this paper, we can prove the Girsanov type theorem under G-framework and the pricing formula. The rest of the paper is organized as follows. In section 2, we review the G-framework established

in [23] and adapt it according to our objective. In section 3, we investigate some properties of

G-expectation by using stochastic control, define the capacity via G-expectation, and show that it is not filtration consistent. In section 4, we investigate the relation between the symmetric

G-martingale and the G-martingale under each probability measure P_v, when the corresponding G-heat

equation is uniformly parabolic. In section 5, based on path wise analysis, we give the martingale characterization for G-Brownian motion. The last section is conclusion and discussion about this work and future work.

2

G-F

RAMEWORK

In this section, we recall the G-framework established in[23]. Let Ω =C0(R+) be the space of all

R-valued continuous paths functions(ωt)t∈R+, withω₀=0. For anyω1,ω2∈Ω, we define

ρ(ω1,ω2):=

∞

X

i=1

2−i[(max

t∈[0,i]|ω

1 t−ω

2 t|)∧1].

We set, for eacht_∈[0,_∞),

W_t := _{η_·∧t :η∈Ω},

Ft := Bt(W) =B(Wt),

Ft+ := Bt+(W) =

\

s>t

Bs(W),

F := _

s>0 Fs.

Then(Ω,_F)is the canonical space with the natural filtration.

This space is used throughout the rest of this paper. For each T >0, consider the following spaces

of ˛aˇrrandom variables ˛a´s

L0_{i p}(_FT):=¦X=ϕ(ω_t1,ω_t2−ω_t1,· · ·,ω_tm−ω_tm

−1),∀m≥1,t1,· · ·,tm∈[0,T],∀ϕ∈l i p(R

m₎©_,

Obviously, it holds that L0_{i p}(_Ft) _⊆ L_{i p}0(_FT), for any t _≤ T < ∞. We notice that X,Y _∈ L0_{i p}(_Ft) meansX Y ∈L0_{i p}(_Ft)and|X| ∈L0_{i p}(_Ft). We further denote

L0_{i p}(_F):=

∞

[

n=1

L0_{i p}(_Fn).

We setB_t(ω) =ωt, and define

E_G[ϕ(B_t+x)] =u(t,x),

whereu(t,x)is the viscosity solution to the following G-heat equation

  

∂u ∂t −G(

∂2u ∂x2) =0, u(0,x) =ϕ(x).

ϕ(_·)_∈l i p(R),G(a) = 1

2_σ2sup

0≤σ2≤1

aσ2,σ0∈[0, 1].

For anyX(ω) =ϕ(Bt1,Bt2−Bt1,· · ·,Btm−Btm−1)∈L

0

i p(F), 0<t1<· · ·<tm<∞, E_G[ϕ(B_t

1,Bt2−Bt1,· · ·,Btm−Btm−1)] =ϕm,

whereϕm is obtained via the backward deduction:

ϕ1(x1,· · ·,xm−1) =EG[ϕ(x1,· · ·,xm−1,Btm−Btm−1)], ϕ2(x1,· · ·,xm−2) =EG[ϕ1(x1,· · ·,xm−2,Btm−1−Btm−2)],

.. .

ϕm−1(x1) =EG[ϕm−2(x1,Bt2−Bt1)], ϕ_m=E_G[ϕ_m−1(Bt1)].

And

EG[X|Ftj] =ϕm−j(Bt1,Bt2−Bt1,· · ·Btj−Btj−1).

Definition 2.1. The expectation E_G[_·], introduced through the above procedure is called G-expectation. The corresponding canonical process B is called a G-Brownian motion under EG[·].

Remark 2.2. As in[23], the G-expectation satisfies

(a) Monotonicity: EG[Y]≥EG[Y′], if Y ≥Y′, Y,Y′∈L0i p(F).

(b) Self dominated property: E_G[X]₋E_G[Y]_≤E_G[X ₋Y], X,Y _∈L0_{i p}(_F). (c) Positive homogeneity E_G[λY] =λE_G[Y],λ≥0, Y _∈L0_{i p}(_F).

(d) Constant translatability EG[X+c] =EG[X] +c, X ∈L0i p(F), c is a constant. Besides(a)_s(d), the conditional G-expectation still satisfies the properties:

(e) Time consistency E_G[E_G[X_|Ft]_|Fs] =E_G[X_|Ft∧s], X_∈L_{i p}0(_F).

(f) E_G[X Y_|Fs] =X+E_G[Y_|Fs] +X−E_G[₋Y_|Fs], X _∈L_{i p}0(_Fs), Y _∈L0_{i p}(_F). (g) E_G[X+Y_|Fs] =X+E_G[Y_|Fs], X _∈L0_{i p}(_Fs), Y _∈L0_{i p}(_F).

Remark 2.3. By properties(c)and(d), E_G[_·]satisfies E_G[c] =c where c is a constant.

Remark 2.4. By theory of stochastic control as in[30], we know that, for any fixed T >0,

E_G[X] = sup

v_·∈Λ′ E

 ϕ

Z t2

t1

v_sd B_s,· · ·,

Z tm

tm−1

v_sd B_s

! = sup

Pv∈Λ

E_Pv[X], (2.1)

where X = ϕ(ωt1,· · ·,ωtm−ωtm−1) ∈ L

0

i p(FT), E is the linear expectation under weiner measure, and G-expectation becomes linear expectation whenσ0=1,{Bt}t≥0is Brownian motion under Weiner

measure. _Z

· 0

vd Bs(·):C[0,T]−→C[0,T]. Here

Λ′ = _{v,v is progressively measurable and quadratic integrable s.t.,

σ2₀≤v2(t)_≤1,a.s. with respect to Weiner measure, 0_≤t≤T}, Λ = _{Pv :Pv is the distribution of

Z ·

0

vsd Bs,v·∈Λ′}.

Lemma 2.5. For any X _∈L0_{i p}(_F), if E_G[_|X_|] =0, then for anyω∈Ω, we have X(ω) =0.

Proof: X =ϕ(ω_t), E_G[X] =sup_v∈Λ′E[|ϕ(

Z t 0

v_sd B_s)_|] =0. Let v ≡1, then E[_|ϕ(B_t)_|] =0. Since

ϕ(_·) is Lipschitz continuous, then ϕ(_·) _≡ 0. Similarly, we can prove that for any X _∈ L0_{i p}(_F)

satisfyingE_G[_|X|] =0, we haveX(ω) =0,∀ω∈Ω.

Remark 2.6. DenotekXk=EG[|X|],∀X∈L0i p(F). By Lemma 2.5, we can prove that(L 0

i p(F),k · k)is a normed space. Let(L1_G(_F),_{k · k})be the completion of(L0_{i p}(_F),_{k · k}), then G-expectation and related conditional expectation can be continuously extended to the Banach space-(L1_G(_F),_{k · k}). G-expectation satisfies properties (a) _∼ (d), and conditional G-expectation satisfies properties (a) _∼ (g). In the completion space, property(f)holds for any bounded random variable X , but we have EG[X Y|Fs] = X E_G[Y],_∀Y _∈L1_G(_FsT),_∀X _∈L1_G(_Fs), and X _≥0. Similarly, we can define_kX_kp=E1/p[_|X_|p],p_≥1. Let L_Gp(_F)be the completion space of L0_{i p}(_F)under normk · kp. Obviously L

p′

G(F)⊂ L p

G(F)for any

1_≤p_≤p′, and it holds for any L_Gp(_Ft)(the proof can be found in[23]).

Definition 2.7. X,Y _∈L_Gp(_FT), X _≤Y in L_Gp, if E_G[((X₋Y)+)p] =0, p_≥1.

Forp_≥1 and 0<T<∞( fixed T). Consider the following type of simple processes

M_Gp,0(0,T) = _{η:ηt(ω) = N_X−1

j=0

ξtj(ω)I[tj,tj+1)(t),

∀N_≥1, 0=t₀<· · ·<t_N=T, ξtj(ω)∈L

p

G(Ftj), j=0.· · ·,N−1}.

For eachηt(ω) =

PN−1

j=0 ξtj(ω)I[tj,tj+1)(t)∈M

p,0

G (0,T), the related Bochner’s integral is defined as

Z T 0

ηt(ω)d t= N_X−1

j=0

ξtj(ω)(tj+1−tj), and

e

E_G[η] = 1

T

Z T 0

E_G[ηt]d t=

1

T N_X−1

j=0

E_G[ξtj(ω)](tj+1−tj).

We can easily check thatEe_G[_·]:M_G1,0(0,T)_→Rsatisfies(a)_∼(d)in Section 2.

kηkp= (

1

T

Z T 0

kηp_tkd t)1/p=

  _T1

N_X−1 j=0

E_G[_|ξtj(ω)|

p_](t

j+1−tj)

  

1/p

.

As discussed in Remark 2.6, k · kp forms a norm inM

p,0

G (0,T). LetM p

G(0,T)be the completion of

M_Gp,0(0,T)under this norm.

3

C

APACITY

U

NDER

G-F

RAMEWORK

3.1

G-Expectation and Related Properties

G-expectation is a kind of time consistent nonlinear expectations, which has the properties of linear expectation in Weiner space except linearity. In this section, we prove some fundamental properties of G-expectation.

We set

Sp=_{M_|M:R+_×Ω_→R,M(t,ω)_∈L_Gp(_Ft),_∀T >0, _{M_{t}t∈[0,T]∈}M_Gp(0,T)_}.

First we will prove that an important class of random variable-bounded continuous functions belongs

to the completion space-L1_G(_F)

Remark 3.1. If X∈L1_G(_FT), there is a sequence fn∈Li p0(FT), such that fnconverges to f in L1G(FT), then for each P_v∈Λ, f_nconverges to f in L1(Ω,P_v), and this convergence is uniform with respect to P_v. We have EG[f] =supPv∈ΛEPv[f](see Proposition 2.2 in[9]).

Lemma 3.2. Λis tight, that is, for anyǫ >0, there exists a compact set K_⊂C([0,T])_⊂Ω, such that for any Pv∈Λ, Pv(Kc)< ǫ, where Kcis the complement of K.

Proof: For any continuous functionx(t), t∈[0,T], define

ωx(δ) = sup

|s−t|≤δ,s,t∈[0,T]|

x_t−x_s|.

By Arzela-Ascoli theorem and Prokhrov theorem (see Theorem 4.4.11 in[4]), to prove the tightness

ofΛ, we only need to prove that for anyα >0, for anyP_v∈Λ, we have lim

δ→0Pv {x :ωx(δ)≥α}

=0. Forα >0, by Proposition 7.2 in the Appendix we know that

lim

δ→0Pv {x :ωx(δ)≥α}

≤ lim

δ→0 E_P

v[ω

2 x(δ)]

α2 =0,

thenΛis tight.

Remark 3.3. For any fixed T, C[0,T]is a polish space, by Prokhrov theorem, we know thatΛis weakly compact. For X_n∈L0_{i p}(_FT), X_n↓0pointwise. As in the Appendix of[9], by Dinni lemma, E_G[X_n]_↓0.

Lemma 3.4. For any fixed T>0,0<t<T , f _∈C_b(W_t), we have f _∈L_G1(_Ft).

Proof: For any bounded continuous f _∈C_b(W_t), _|f_{| ≤}M, M>0, there exists a sequence of random

variables fn∈L0i p(Ft), such thatfnmonotonically converges to f.

AsΛis tight, for anyǫ >0, there exists a compact setK_{∈ FT}, such that

sup

Pv∈Λ

EPv[IKc]< ǫ,

whereKcis the complement ofK. Since a compact set is closed in any metric space, we know that

Kcis an open set, hence

EG[IKc] =sup

Pv∈Λ

EPv[IKc]< ǫ.

And by Dini’s theorem on any compact setK, fnconverges to f uniformly,

lim

n→∞EG[|fn−f|]≤nlim→∞[EG[|fn−f|IK] +EG[|fn−f|IKc]]≤Mǫ,

then lim_n→∞E_G[_|f_n−f|] =0, note that this convergence is uniform with respect to t, sof ∈L1_G(_FT).

3.2

Capacity under G-Framework

Since the publication of Kolomogrov’s famous book on probability, the study of “the nonlinear

prob-ability" theory named “capacity" has been studied intensively in the past decades, see [5], [8],

[12], [16], [18],[22],[26]. Hence it is meaningful to extend such a theory to G-framework, and

this section contributes to such an extension. We shall now define a nonlinear measure through G-expectation and investigate its properties.

Definition 3.5. PG(A) = supPv∈ΛPv(A), for any Borel set A, where Pv is the distribution of

Z t 0

vd Bs and v is a bounded adapted process, andΛis the collection of all such Pv.

By Remark 3.1 and Remark 3.3, PG is a regular Choquet capacity (we call it G-capacity), that is, it

has the following properties:

(1) For any Borel setA, 0_≤P_G(A)_≤1;

(2) IfA_⊂B, then P_G(A)_≤P_G(B);

(3) IfAn is a sequence of Borel sets, thenPG(

S

nAn)≤

P

nPG(An);

(4) IfA_n is an increasing sequence of Borel sets, then

P_G([

n

A_n) =lim

n PG(An).

Remark 3.6. Here property (4) dose not hold for the intersection of decreasing sets. There are two ways to define capacity under G framework, see[10].

LetΛebe the closure ofΛunder weak topology. (1) PG(A) =supPv∈ΛPv(A),

(2) P_G(A) =sup_P∈ΛeP_v(A).

Then P_Gand P_Gall satisfy the properties in Remark 3.2.

We use the standard capacity related vocabulary: A set A is polar if P_G(A) =0, a property holds ˛ aˇrquasi-surely ˛a´s (q.s.), if it holds outside a polar set. Here P_Gquasi-surely is equivalent to P_Gquasi-surely. Even in general, PG(A)≤ PG(A), but if PG(A) =0, IA∈ L1G(FT). Then by Theorem 59 in [10](page 24), PG(A) =PG(A) =0. Thus, a property holds PG-quasi-surely if and only if it holds PG-quasi-surely.

Remark 3.7. As in the Appendix of [9], we consider the Lebesgue extension of G-expectation, we can define G-expectation for a large class of measurable functions, such as all the functions with

sup_P

v∈ΛEPv[|X|] < ∞, but we can not define G conditional expectation. So far we can only define

G conditional expectation for the random variables in L1_G(_F), in the rest of the paper, we denote

sup_P

v∈ΛEPv[X] =EG[X], but that dose not mean X ∈L

1 G(F).

First we give the property of this Choquet capacity-PG:

Proposition 3.8. Let p_≥1.

(1) If A is a polar set, then for anyξ∈Lp_G(_FT), E_G[I_Aξ] =0. (2) P_G{|ξ|>a_{} ≤} EG[|ξ|

p_] ap ,ξ∈L

p

G(FT), a>0.

(3) If X_n, X _∈ L_Gp(_F), E_G[_|X_n−X_p|p]_→0, then there exists a sub-sequence X_n

k of Xn, such that

Proof: (1)Without loss of generality, let p=1. Ifξ∈L0_{i p}(_FT), then E_G[I_Aξ] =0. Ifξ∈L1_G(_FT),

then there exists a sequence of random variables ξn, which satisfies EG[|ξn −ξ|] −→ 0, and

E_G[_|ξ_nI_A−ξI_A|]_−→0, hence we obtain

E_G[ξI_A] = lim

n→∞EG[ξnIA] =0.

(2) From

EG[|ξ|p] =EG[|ξ|pI{|ξ|>a}+|ξ|pI{|ξ|≤a}]≥apPG{|ξ|>a},

we get the result.

(3) By (2), we know that for anyǫ >0,

lim

n→∞PG{|Xn−X|> ǫ}=0.

Then for every positive integer k, there existsn_k>0, such that

P_G{|X_n−X_{| ≥} 1

2k}< 1

2k, ∀n≥nk.

Supposen₁<n₂<. . .<n_k<. . ., letX′_k=X_nk be a sub-sequence ofX_n. Then

P_G{X′_k9X}=PG

( [

m

\

k

[

v

|X′_k+v−X| ≥ǫ_m

)

≤ X

m P_G

( \

k

[

v

|X_k′_+v−X_{| ≥}ǫm

)

≤ X

m P_G

( [

v

|X_k′

0+m+v−X| ≥ǫm

)

≤ X

m

X

v P_G

|X′_k

0+m+v−X| ≥

1 2k0+m+v

≤ X

m

X

v

1 2k0+m+v

= 1

2k0 →0.

Therefore,P_G{X_k′ ₉X_}=0.

Next we investigate the relation betweenX _≤Y in L_Gp, andX_≤Y, q.s.

Lemma 3.9. E_G[(X₋Y)+] =0if and only if X_≤Y , q.s.

Proof: Without loss of generality, suppose thatp=1, ifX ≤Y, q.s.,{X−Y ≥0}is a polar set. Then by Proposition 3.8, we know that

E_G[(X₋Y)+] =E_G[(X₋Y)I_{X−Y≥0}] =0.

IfX _≤Y inL1_G, which means E_G[(X₋Y)+] =0, then by Proposition 3.8, for anyǫ >0, we have

PG[(X−Y)+> ǫ]≤

E_G[(X₋Y)+]

Letǫ↓0, forP_Gis a Choquet capacity, we know

P_G[(X₋Y)+>0] =lim

ǫ→0PG[(X−Y)

+_{> ǫ] =0,}

so we haveP_G[X _≥Y] =P_G[(X₋Y)+>0] =0, that isX _≤Y q.s.

Remark 3.10. After defining the capacity, one important issue is whether I_A∈ L_G1(_F), for any Borel set A_{∈ F}. Here we give a counter example that there exists a sequence of Borel sets which do not belong to L1_G(_F).

Example 3.11. Let

A_n=

 

ω: limt→0

Bt+s−Bs

p

2tlog log 1/t ∈

(1₋1/2n, 1)

 

, n=1, 2,· · ·

Then A_n↓φ, for any fixed n, let v_≡1₋ 1

3n, Ôò

P_v(A_n) =P

 

ω:(1−

1

3n)limt→0

Bt+s−Bs

p

2tlog log 1/t ∈

(1₋1/2n, 1)

  =1,

which meanslim_n→∞P_G(A_n) =1.

For any sequences of random variables{X_n} ⊂ L1_G(_F), satisfying X_n ↓0 q.s., we have E_G[X_n]_↓0,

see Theorem 26 in[10]. So the setsA_ndo not belong to L1_G(_F).

Actually, the next lemma tells us even not all the open Borel sets belong toL1_G(_F).

Lemma 3.12. There exists an open set A_{∈ FT}, such that I_Adose not belongs to L1_G(_FT).

Proof: We prove this result by contradiction. If for any open setA_{∈ FT}, I_A∈ L1_G(_FT) holds. For

all the open sets satisfyingAn ↓φ, we have PG(An) ↓0. Because for any Borel setB, there exists

compact sets_{F_{n} ⊂}B satisfyingP_G(B_\F_n)_↓0, see[16]. Then we have for any Borel setB_{∈ FT},

I_{B ∈} L_G1(_FT). But Example 3.11 shows that not all the Borel sets belong to L1_G(_FT), which is a contradiction.

Then we can not define conditional G-expectation for I_A, where Ais any Borel set, and even for

any open set, that means we can not define conditional G-capacity, and that is why we claim that G-expectation is not filtration consistent.

4

S

YMMETRIC

M

ARTINGALE IN

G-F

RAMEWORK

We begin with the definition of martingale in G-framework.

Definition 4.1. M ∈ S2 is called a martingale, if for any0≤ s ≤ t < ∞, it satisfies E[Mt|Fs] = M_s; if furthermore M is symmetric, that is E[₋M_t|Fs] =₋E[M_t|Fs], then it is called a symmetric martingale.

In this section, when the corresponding G-heat equation is uniformly parabolic, which meansσ0>

0, we prove that the symmetric martingale is a martingale under each probability measurePv, and

4.1

Path analysis

In this section, we give some path properties of quadratic variation process _〈B_〉t and the related

stochastic integral

Z t 0

η(s)d B_s,η∈M_G2(0,T)and

Z t 0

η1(s)d〈B〉s,η1(s)∈MG1(0,T).

From the definition of E_G[_·], we know that the canonical processB_t is a quadratic integrable

mar-tingale under eachP_v. So they have a universal version of “quadratic variation process ofB_t", and

by the definition of stochastic integral with respect to G-Brownian motion, for any η∈ M_G2(0,T),

Z T 0

ηsd Bs is well defined, which means

Z T 0

ηsd Bs is aPv local martingale. Similar arguments can

be found in Lemma 2.10 in [10]. Then, due to the Doob’s martingale inequality for each P_v, the

following result holds.

Lemma 4.2. For anyη∈M_G2(0,T),

Z t 0

η(s)d B_s is quasi-sure continuous.

Proof:Ifη∈M_G2,0(0,T), then the result is true. Ifη∈M_G2(0,T), then there exist_{ηn} ⊂MG2,0(0,T)

such that

Z T 0

EG[|η(s)−ηn(s)|2]ds−→0.

For we have

sup

Pv∈Λ

E_Pv[ sup

0≤t≤T|

Z t 0

(η−η_n)d B_s|2]_≤Ksup

Pv∈Λ

E_Pv[

Z T 0

(η−η_n)2d〈B〉s]

≤ K sup

Pv∈Λ

EPv[

Z T 0

(η−ηn)2ds]≤K

Z T 0

sup

Pv∈Λ

EPv[(η−ηn)

2_]ds −→0.

Hence,

Z t 0

ηn(s)d Bs uniformly converges to

Z t 0

η(s)d B_s q.s. Therefore,

Z t 0

η(s)d B_s is continuous

q.s.

By similar argument we can get the following lemma.

Lemma 4.3. For anyη∈M_G1(0,T),

Z t 0

η(s)d〈B〉s and

Z t 0

η(s)ds are quasi-surely continuous.

4.2

Representation theorem

In this section, we are concerned with the G-martingale when the corresponding G-heat equation

is uniformly parabolic (σ0 > 0). In the following, for any X ∈ L0i p(FT), we will give X a

repre-sentation in terms of stochastic integral. For this part, Peng gives the conjecture for reprerepre-sentation

theorem of G-martingales. Soner et al[21]prove this theorem for a large class of G-martingales by

BSDE method. Here for the ease of exposition, we prove the theorem separately for some special martingales.

Lemma 4.4. Supposeσ0>0, if u is the solution of G-heat equation, then we have

u(r,B_t−r) =u(t, 0) +

Z t−r

0

u_x(B_v)d B_v+

Z t−r

0

u_{x x}(B_v)d_〈B_〉v−

Z t−r

0

(u+_{x x−}σ2₀u−_{x x})d v.

Proof:The proof follows that of Itô’s formula. Due to the regularity of parabolic equation, see[18],

we knowu,u_x,u_{x x} are all uniformly continuous. Since Lipschitz continuous functions are dense in

uniform continuous functions, we assume thatu,u_x,u_{x x} are Lipschitz continuous.

Letδn=

t−r

n , we have

u(r,B_t−r)₋u(t, 0) =

n−1

X

k=0

”

u(t₋(k+1)δn,B(k+1)δn)−u(t−kδn,Bkδn)

—

=

n−1

X

k=0

”

u(t−(k+1)δn,B(k+1)δn)−u(t−kδn,B(k+1)δn)

—

+

n−1

X

k=0

”

u(t₋kδn,B(k+1)δn)−u(t−kδn,Bkδn)

—

=

n−1

X

k=0

”

−u_t(t₋kδn,Bkδn)δn+ux(t−kδn,Bkδn)(B(k+1)δn−Bkδn) +1

2ux x(t−kδn,Bkδn)(B(k+1)δn−Bkδn)

2

−ξn+ηn,

where

ηn =

1 2

n−1

X

k=0

[u_{x x}(t₋kδn,Bkδn+θ1(B(k+1)δn−Bkδn))−ux x(t−kδn,Bkδn)](B(k+1)δn−Bkδn)

2_,

ξn = n−1

X

k=0

”

ut(t−kδn+θ2δn,B(k+1)δn)−ut(t−kδn,B(k+1)δn)

—

δn,

+

n−1

X

k=0

”

u_t(t₋kδn,B(k+1)δn)−ut(t−kδn,Bkδn)

—

δn,

andθ1,θ2are constants in[0, 1], which depend onω, tandn. Hence,

E_G[_|η_n|] _≤

n−1

X

k=0

E_G”|u_{x x}(t−kδ_n,B_kδn+θ(B(k+1)δn−Bkδn))

−u_{x x}(t₋kδn,Bkδn)|(B(k+1)δn−Bkδn)

2— ≤ K

n−1

X

k=0

E_G[_|B_(k+1)δ

n−Bkδn|

3_] ≤K

n−1

X

k=0

δ3/2_n −→0.

Then, we have

EG[| n

X

k=1

−ut(t−kδn,Bkδn)I[kδn,(k+1)δn)(v)−ut(t−v,Bv)|]≤C(δn+δ

1/2

n )−→0.

Therefore,

n−1

X

k=0

−u_t(t₋kδn,Bkδn)δn−→

Z t−r

0

−u_t(t₋v,B_v)d v. Similarly we get

n−1

X

k=0

u_x(t₋kδn,Bkδn)(B(k+1)δn−Bkδn)−→

Z t−r

0

u_x(t₋v,B_v)d B_v. Note

E_G[_|

n−1

X

k=0

u_{x x}(t₋kδn,Bkδ_n)(B(k+1)δ_n−Bkδ_n)2− n−1

X

k=0

u_{x x}(t₋kδn,Bkδ_n)(〈B〉(k+1)δ_n− 〈B〉kδ_n)|2]

≤

n−1

X

k=0

E_G[_|u_{x x}(t₋kδn,Bkδn)|

2

|(B_(k+1)δ

n−Bkδn)

2

−(_〈B_〉(k+1)δ

n− 〈B〉kδn)|

2_]

+ 2X

j6=i

E_G[u_{x x}(t₋jδn,Bjδn)ux x(t−iδn,Biδn)((B(j+1)δn−Bjδn)

2

−(_〈B_〉(j+1)δn− 〈B〉jδn)) ((B(i+1)δn−Biδn)

2

−(_〈B〉(i+1)δn− 〈B〉iδn))]

≤

n−1

X

k=0

E_G[(c+c|B_kδn|

2₎ |

Z (k+1)δn

kδn

(B_v−B_kδn)d Bv|

2_]

≤

n−1

X

k=0 C EG[|

Z (k+1)δn

kδn

(Bv−Bkδn)d Bv|

2_] ≤C

n−1

X

k=0

δ2_n−→0, and

n−1

X

k=0

u_{x x}(t₋kδn,Bkδn)(〈B〉(k+1)δn− 〈B〉kδn)−→

Z t−r

0

u_{x x}(t₋v,B_v)d_〈B_〉v.

Sinceusolves G-heat equation, we get

u(r,B_t−r)₋u(t, 0) =

Z t−r

0

−u_t(t−v,B_v)d v+

Z t−r

0

u_x(t−v,B_v)d B_v

+

Z t−r

0

ux x(t−v,Bv)d〈B〉v

=

Z t−r

0

ux(t−v,Bv)d Bv+

Z t−r

0

ux x(t−v,Bv)d〈B〉v

−1

2

Z t−r

0

Theorem 4.5. Whenσ0>0, then for any X ∈Li p0(FT), we have

X =EG[|ϕ(·)|] +

Z tm

0

Zsd Bs+

Z tm

0

η(s)d〈B〉s−

Z tm

0

(η+−σ2₀η−)ds.

Proof:Whenm=1, for the regularity of theu, lim_r→0u(r,B_t−r(ω)) =ϕ(B_t(ω)). By Lemma 4.4, and path analysis in Section 3, we know

ϕ(Bt) = u(t, 0) +

Z t 0

ux(t−v,Bv)d Bv

+

Z t 0

u_{x x}(t₋v,B_v)d_〈B_〉v−

Z t 0

(u+_{x x−}σ2₀u−_{x x})d v.

Due to the definition of G-expectation,EG[ϕ(Bt)] =u(t, 0), so the result holds whenm=1.

By similar argument, we get

ϕ(B_T−B_t) = u(T₋t, 0) +

Z T t

u_x(T₋v,B_v−B_t)d B_v

+

Z T t

u_{x x}(T−v,B_v−B_t)d〈B〉v−

Z T t

(u+_{x x}−σ2₀u−_{x x})d v.

Whenm=2, for each x

ϕ(x,BT −Bt) = u(T−t,x, 0) +

Z T t

uy(T−v,x,Bv−Bt)d Bv

+

Z T t

u_{y y}(T₋v,x,B_v−B_t)d_〈B_〉v

−

Z T t

(u+_{y y}(T−v,x,Bv−Bt)−σ20u−y y(T−v,x,Bv−Bt))d v.

By continuous dependence estimate theorem in [13], we know for each fixed t, u(T ₋t,x, 0) is

lipschitz continuous and bounded with respect tox, then there existη∈M_G1(0,T)andz∈M_G2(0,T),

such that

u(T₋t,B_t, 0) =E_G[_|u(T₋t,B_t, 0)_|] +

Z t 0

z_sd B_s+

Z t 0

ηsd〈B〉s−

Z t 0

(η+_s −σ2₀η−_s )ds. (4.2)

That is

ϕ(B_t,B_{T −}B_t) =E_G[ϕ(B_t,B_{T −}B_t)] +

Z T 0

z_sd B_s+

Z T 0

ηsd〈B〉s−

Z T 0

(η+_s −σ2₀η−_s )ds.

Hereηandzare different from (4.2).

4.3

Properties for the Symmetric Martingale

From[23], we know E_G[(_〈B_〉t−t)+] =E_G[(σ2₀t_{− 〈}B_〉t)+] =0. Thenσ2₀t _{≤ 〈}B_{〉t ≤}t, andσ2₀(T₋

t)_{≤ 〈}B_{〉T − 〈}B_{〉t ≤}T₋t, for anyt _≤T.

Lemma 4.6. For anyη∈M_G1(0,T), we have

Z T 0

η+(s)d〈B〉s−

Z T 0

η+(s)ds≤0q.s.,and

Z T 0

η−(s)d〈B〉s−σ02

Z T 0

η−(s)ds≥0.

Proof:Ifη∈M_G1,0(0,T), then

Z T 0

η+(s)d〈B〉s= n−1

X

j=0

ξ+_j(_〈B〉tj+1− 〈B〉tj)≤

n−1

X

j=0

ξ+_j(t_j+1−tj) =

Z T 0

η+(s)ds,

and _Z

T

0

η−(s)d_〈B_〉s=

n−1

X

j=0

ξ−_j (_〈B_〉t

j+1− 〈B〉tj)≥σ

2 0

n−1

X

j=0

ξ+_j(t_j+1−t_j) =σ2₀

Z T 0

η−(s)ds.

Ifη∈M_G1(0,T), then there existsη_n∈M_G1,0(0,T), such that

E_G

 

Z T 0

(ηn−η)d〈B〉s



 _−→ 0,

EG

 

Z T 0

(ηn−η)ds



 _−→ 0.

Then there exists a subsequenceηk⊂ηn such that

Z T 0

(ηk−η)d〈B〉s−→0, q.s.,

and _Z

T

0

(ηk−η)ds−→0, q.s..

Therefore

Z T 0

η+(s)d〈B〉s−

Z T 0

η+(s)ds ≤ 0, q.s.

Z T 0

η−(s)d_〈B_〉s−σ₀2

Z T 0

η−(s)ds _≥ 0, q.s.

Theorem 4.7. Supposeσ0>0, then for any X∈L1G(FT),η=EG[X | Ft]satisfies

sup

Pv∈Λ

Proof:IfX _∈L0_{i p}(_FT), then

X_T =E_G[X_T] +

Z T 0

z_sd B_s+

Z T 0

ηsd〈B〉s−

Z T 0

(η+−σ₀2η−)ds.

By Theorem 4.1.42 in[24], we have

η=E_G[X_T | Ft] =EG[XT] +

Z t 0

z_sd B_s+

Z t 0

η_sd〈B〉s−

Z t 0

(η+−σ2₀η−)ds.

Since

Z t 0

z_sd B_s is a quadratic integrable martingale for eachP_v, then

sup

Pv∈Λ

E_Pv[(X−η)I_A] = sup

Pv∈Λ

E_Pv

 (

Z T t

η_sd〈B〉s−

Z T t

(η+−σ2₀η−)ds+

Z T t

z_sd B_s)I_A

 

= sup

Pv∈Λ

EPv

 (

Z T t

ηsd〈B〉s−

Z T t

(η+−σ2₀η−)ds)IA

 _≤0. On the other hand

sup

Pv∈Λ

EPv

 (

Z T t

ηsd〈B〉s−

Z T t

(η+−σ2₀η−)ds)IA

 _≥ sup

Pv∈Λ

EPv

 

Z T t

ηsd〈B〉s−

Z T t

(η+−σ2₀η−)ds

 =0.

Therefore sup_P

v∈ΛEPv[(X−η)IA] =0.

IfX _∈L_G1(_FT), then there exist X_n∈L0_{i p}(_FT), such thatE_G[_|X₋X_n|]_→0 and

E_G[_|E_G[X _{| Ft}]₋E_G[X_{n| Ft}]_|]_→0. Hence,

sup

Pv∈Λ

E_P

v[(X−EG[X | Ft])IA] = sup

Pv∈Λ

E_P

v[(X−Xn+EG[Xn| Ft]−EG[X | Ft])IA + (X_n−E_G[X_{n| Ft}])I_A]

≤ E_G[_|X₋X_n|] +E_G[_|E_G[X _{| Ft}]₋E_G[X_{n| Ft}]_|]

+ sup

Pv∈Λ

E_Pv[(X_n−E_G[X_{n| Ft}])I_A]_−→0. On the other hand

sup

Pv∈Λ

E_Pv[(X−E_G[X | Ft])IA] ≥ sup Pv∈Λ

E_Pv[(X_n−E_G[X_n| Ft])IA]

−E_G[_|X−X_n|]₋E_G[_|E_G[X| Ft]−EG[Xn| Ft]|]−→0.

So sup_P

v∈ΛEPv[(X−EG[X| Ft])IA] =0.

The next corollary states that there is a universal version of the conditional expectation for the

Corollary 4.8. Ifσ0>0, MT ∈LG1(FT), and−EG[MT | Ft] =EG[−MT | Ft], then E_G[M_{T | Ft}] =E_Pv[M_{T | Ft}], P_v.a.s.

Then consequently we have the main result of this section.

Theorem 4.9. If M is a symmetric martingale under G expectation withσ0>0, then M is a martingale under each Pv.

Remark 4.10. We can not hope that

E_Pv[X_|Ft]_≤E_G[X_|Ft] =₋E_G[₋X_|Ft]_{≤ −}E_Pv[₋X_|Ft] =E_Pv[X_|Ft]

Obviously for any X ∈L1_G(_F), EG[X]≥ EPv[X]. But this is not trivial for conditional G expectation,

since it is defined as the “Markovian" version, so E_P

v[X|Ft]≤EG[X|Ft]does not hold for each Pv. As a corollary, we can get the Doob’s Martingale inequality for “symmetric G-martingale".

Corollary 4.11. When σ0 > 0, ff MT ∈ L2G(FT), and−EG[MT | Ft] = EG[−MT | Ft]and Mt is quasi-surely continuous then we have PG[sup0≤t≤T|Mt| ≥λ]≤

1

λpEG[|MT|

p_{], for any p}

≥1, T ≥0, as a special case EG[sup0≤t≤T|Mt|2]≤2EG[|MT|2].

Proof: By the Doob’s martingale inequality in probability space, we have

P_G[ sup

0≤t≤T|

M_{t| ≥}λ]

= sup

Pv∈Λ

P_v[ sup

0≤t≤T|

M_{t| ≥}λ]

≤ sup

Pv∈Λ 1

λpEPv[|MT|

p_]

= 1

λpEG[|MT|p].

5

M

ARTINGALE

C

HARACTERIZATION OF

G-B

ROWNIAN

M

OTION

We are ready to prove the Martingale Characterization theorem for G-Brownian Motion.

Theorem 5.1. Let M _{∈ S}2. Suppose that (I) M is a symmetric martingale; (II) M_t2₋t is a martingale;

(III) forσ0>0,σ20t−Mt2 is a martingale; and

Then M is a G-Brownian motion in the sense that M has the same finite distribution as the G-Brownian motion B.

Remark 5.2. The Lèvy’s martingale characterization of Brownian motion in a probability space states that, Bt is a continuous martingale with respect to Ft, and B2t − t is a Ft martingale, iff Bt is a Brownian motion. Our martingale characterization of G-Brownian motion covers Lèvy’s martingale characterization of Browninan motion, when σ0 = 1. In a probability space, the quadratic process 〈B〉t of Brownian motion Bt almost sure equals to t with respect to the probability measure P. But in the G-framework, the quadratic process_〈B_〉t of G -Brownian motion B_t is not a fixed function any more. Instead, it is a stochastic process in G-framework. The criteria(I I I) is the description of the nonsymmetric property for 〈B〉t. Condition (IV) is reasonable, thanks to [10], for any G-Brownian motion, it has continuous path.

In a probability space, thanks to the characteristic function of normal distribution, Lèvy’s martingale characteristic theorem of Brownian motion holds. But in G-framework, there is no characteristic

function of “G-normal distribution"(see the definition in[23]), we have to find a different method

to solve this problem.

Next we shall prove Theorem 5.1 in 4 steps.

Proof of Theorem 5.1:

Step 1. _∀η ∈ M_G2(0,T), we define the stochastic integral of Itô’s type

Z T 0

ηd M_t, and prove that

Z t 0

ηd M_sis a symmetric martingale underE_G[_·].

By conditions(I)_∼(I V)in Theorem 5.1, we have the following proposition.

Proposition 5.3. for any tj≤tj+1<ti≤ti+1, (1) E_G[(M_t

j+1−Mtj)ξtj(Mti+1−Mti)ξti] =0,ξtj ∈L

p

G(Ftj),ξti ∈L

p G(Fti),

(2) E_G[(M_t

j+1−Mtj)

2

|Ftj] =EG[(Mtj+1−Mtj)

2_{] =t}

j+1−tj, (3) E_G[X+ξtj(Mtj+1−Mtj)

2_{] =E}

G[X+ξtj((Mtj+1)

2 −(M_t

j)

2_{)], X}

∈L_Gp(_FT),ξtj∈L

p G(Ftj).

Let

Z t 0

ηd M_s=

Z T 0

ηI[0,t](s)d Ms. The next lemma shows that

Z t 0

ηd M_s is a symmetric martingale.

Lemma 5.4. For anyη∈M_G2(0,T),t_∈[0,T],

Z t 0

ηd M_s is a symmetric martingale.

Step 2: The quadratic variation process _〈M_〉t of M_t exists. Defining the stochastic integral with

respect to〈M〉tin MG1(0,T), we can get the isometric formula in the G-framework.

Now we give the isometry formula in the G-framework. First, we give a proposition.

Proposition 5.5. For all0_≤s_≤t,ξ∈L1_G(_Fs)and X _∈L1_G(_F)we have

E_G[X+ξ(M_t2₋M_s2)] =E_G[X+ξ(M_t−M_s)2] =E_G[X+ξ(_〈M_{〉t− 〈}M_〉s)].

Lemma 5.6. Forη∈M_G2(0,T), it holds that

E_G[(

Z T 0

η(s)d M_s)2] =E_G[

Z T 0

η2(s)d_〈M_〉s]_≤

Z T 0

E_G[η2(s)]ds.

Remark 5.7. We can prove Lemma 5.4 and 5.6 by the similar method to that in[23]and[29]. Thus we omit the proof.

Next, we investigate the property of the quadratic process_〈M_〉t.

Lemma 5.8. (1)

E_G

 

‚Z t s

(M_v−M_s)d M_v

Œ2

Fs

 =E_G

–Z t s

(M_v−M_s)2d_〈M_〉v

™

= 1

2(t−s)

2_.

(2) For anyη∈M_G1(0,T), 1

2

Z t 0

ηsd〈M〉s−

Z t 0

G(ηs)ds is a martingale.

Proof: (1) Lets=t₀<t₁<. . .<t_N =t, andt_j+1−t_j= t−s

N . Note that

N_X−1 j=0

(Mtj−Ms)(Mtj+1−Mtj)−→

Z t s

(Mv−Ms)d Mv in L2G(Ft).

Then _

 

N_X−1 j=0

(M_tj−M_s)(M_tj+1−M_tj)

   2 −→ ‚Z t s

(M_v−M_s)d M_v

Œ2

in L1_G.

And we have

EG     EG   Z t s

(Mv−Ms)d Mv

Œ2

Fs

 ₋EG

      

N_X−1 j=0

(Mtj−Ms)(Mtj+1−Mtj)

   2 Fs        

≤ E_G

   EG

    ‚Z t s

(M_v−M_s)d M_v

Œ2 −

  

N_X−1 j=0

(M_tj−M_s)(M_tj+1−M_tj)

   2 Fs        

= E_G

    ‚Z t s

(M_v−M_s)d M_v

Œ2 −

  

N_X−1 j=0

(M_t

j−Ms)(Mtj+1−Mtj)

   2    −→0.

So EG       

N_X−1 j=0

(Mtj−Ms)(Mtj+1−Mtj)

   2 Fs   

−→EG

 

Z t s

(Mv−Ms)d Mv

Œ2

Fs

  in L1_G.

Now EG       

N_X−1 j=0

(Mtj−Ms)(Mtj+1−Mtj)

   2 Fs    

= E_G

      

N_X−1 j=0

(M_tj−M_s)2(M_tj+1−M_tj)

   2 Fs     =

N_X−1 j=0

(t_j−s)(t_j+1−t_j).

We know E_G   Z t s

(M_v−M_s)d M_v

Œ2

Fs

 = lim

N→∞

N_X−1 j=0

(t_j−s)(t_j+1−t_j) =

Z t s

vd v= 1 2(t−s)

2_.

(2) For any η ∈ M_G1,0(0,T), ηt =

PN−1

j=0 ξtj(ω)I[tj,tj+1)(t), for any t ∈[0,T], suppose t = tN−1.

Then, E_G   Z T 0

ηsd〈M〉s−2

Z T 0

G(ηs)ds

Ft  

= E_G

  

N_X−1 j=0

ξj(〈M〉tj+1− 〈M〉tj)−2

N_X−1 j=0

G(ξj)(tj+1−tj)

FtN−1

  

=

N_X−2 j=0

ξj(〈M〉tj+1− 〈M〉tj)−2

N_X−2 j=0

G(ξj)(tj+1−tj) +ξ+tN−1EG

•

〈M〉tN− 〈M〉tN−1

FtN−1

˜

+ξ−_t

N−1EG

•

−(_〈M_〉t

N− 〈M〉tN−1)

FtN−1

˜

−2G(ξN−1)(tN−tN−1)

=

Z t 0

ηsd〈M〉s−2

Z t 0

G(ηs)ds+2G(ξN−1)(tN−tN−1)−2G(ξN−1)(tN−tN−1)

=

Z t 0

ηsd〈M〉s−2

Z t 0

G(ηs)ds.

Ifη∈M_G1(0,T), then there exists a sequenceηN ∈M_G1,0(0,T), such thatηN −→ηin M_G1(0,T), and (ηN₎+

and

u(t,Mt+δ−Mδ)−u(s,Ms+δ−Mδ) =

N_X−1 j=0

h

u(t_j+1,M_δ+t

j+1−Mδ)−u(tj,Mtj+1+δ−Mδ)

+u(t_j,Mδ+tj+1−Mδ)−u(tj,Mtj+δ−Mδ)

i

=

N_X−1 j=0

€

u(vj+1−δ,Mv−Mδ)−u(vj−δ,Mv−Mδ) Š

+

N_X−1 j=0

ux(vj−δ,Mvj−Mδ)(Mvj+1−Mvj)

+1

2 N_X−1

j=0

ux x(vj−δ,Mvj−Mδ)(Mvj+1−Mvj)

2

+ 1

2 N_X−1

j=0

h

u_{x x}(v_j−δ,M_v

j−Mδ+θj(ω)(Mvj+1−Mvj))−ux x(vj−δ,Mvj−Mδ)

i (M_v

j+1−Mvj)

2

=

N_X−1 j=0

Z vj+1

vj

u_t(v₋δ,M_v

j+1−Mδ)d v+

N_X−1 j=0

u_x(v_j−δ,M_v

j −Mδ)(Mvj+1−Mvj)

+1

2 N−_X1

j=0

ux x(vj−δ,Mvj −Mδ)(Mvj+1−Mvj)

2₊

1 2

N_X−1 j=0

h

u_{x x}(v_j−δ,M_vj−Mδ+θj(ω)(Mvj+1−Mvj))−ux x(vj−δ,Mvj−Mδ)

i

(M_vj+1−M_vj)2, whereθj(ω)∈(0, 1), vj=δ+tj.

Now that E_G   

N_X−1 j=0

Z vj+1

vj

u_t(v₋δ,M_v

j−Mδ)d v−

Z t+δ

s+δ

u_t(v₋δ,M_v−Mδ)d v   

≤ E_G

  

N_X−1 j=0

Z vj+1

vj

C_|M_v−M_v

j|d v

   ≤ N_X−1

j=0 2

3|vj+1−vj| 3/2₌

N_X−1 j=0

2 3δ

3/2 N −→0, and

N_X−1 j=0

u_x(v_j−δ,M_v

j−Mδ)I[tj,tj+1)−→ux(v−δ,Mv−Mδ),

we have N_X−1

u_x(v_j−δ,M_v

j−Mδ)(Mvj+1−Mvj)−→

Z t+δ

Since E_G    

N_X−1 j=0

u_{x x}(v_j−δ,M_vj−Mδ)(Mvj+1−Mvj)

2 −

N_X−1 j=0

u_{x x}(v_j−δ,M_vj−Mδ)(〈M〉vj+1− 〈M〉vj+1) 2   

= E_G

    N−1_X j=0

u_{x x}(v_j−δ,M_vj −Mδ) Z vj+1

vj

(M_v−M_vj)d M_v

2   

≤ E_G

  

N_X−1 j=0

u2_{x x}(v_j−δ,M_vj−Mδ)  

Z vj+1

vj

(M_v−M_vj)d M_v

 

2

 +2E_G

  

X

j6=i

u_{x x}(v_j−δ,M_vj−Mδ)

ux x(vi−δ,Mvi −Mδ)

Z vj+1

vj

(Mv−Mvj)d Mv

Z vi+1

vi

(Mv−Mvi)d Mv

 

≤ N_X−1

j=0

E_G[u2_{x x}(v_j−δ,M_v

j)−Mδ]δ

2 N ≤C

N_X−1 j=0

E_G[1+δ+_|M_v

j−Mδ|]δ

2

N −→0, we have

N_X−1 j=0

u_{x x}(v_j−δ,M_vj−Mδ)(〈M〉vj+1− 〈M〉vj)−→

Z t+δ

s+δ

u_{x x}(v₋δ,M_v−Mδ)d〈M〉v.

So the result holds.

Step 4.

Lemma 5.11. If u(t,x)is the viscosity solution to the G-heat equation, andσ0>0, then EG[ϕ(Mt+x)] =u(t,x), ϕ∈l i p(R).

Proof: For 0< σ0≤1, then by Theorem 4.13 in[17],u(t,x)∈C1,2((0,T]×R), andu(t,x),ut(t,x) andu_{x x}(t,x)are all uniformly continuous,u_x(t,x) is bounded and continuous, see in[28]. Then by Lemma 5.10, for any 0< ǫ <v≤T, we have

u(ǫ,Mδ+v−ǫ−Mδ+x)−u(v,x) =

Z v−ǫ+δ δ

−ut(v+δ−s,Ms−Mδ+x)ds

+

Z v−ǫ+δ δ

u_x(v+δ−s,M_s−Mδ+x)d Ms+ 1 2

Z v−ǫ+δ δ

By Lemma 5.8, it is easy to check that

EG[u(ǫ,Mδ+v−ǫ−Mδ+x)]−u(v,x)

= E_G

 

Z v−ǫ+δ δ

−u_t(v+δ−s,M_s−M_δ+x)ds

+ 1

2

Z v−ǫ+δ δ

u_{x x}(v+δ−s,M_s−Mδ+x)d〈M〉s  

= EG

 1

2

Z v−ǫ+δ δ

ux x(v+δ−s,Ms−Mδ+x)d〈M〉s

−

Z v−ǫ+δ δ

G(u_{x x}(v+δ−s,M_s−M_δ+x))ds

 

= 0.

Then limǫ→0EG[u(ǫ,Mδ+v−ǫ−Mδ+x)] =u(v,x).

We also have

EG[|u(ǫ,Mδ+v−ǫ−Mδ+x)−u(0,Mδ+v−Mδ+x)]≤2C EG[ p

ǫ+_|Mδ+v−ǫ−Mδ+v|]≤4C

p

ǫ, whereC is the Lipschitz constant ofϕ.

ThenE_G[ϕ(Mδ+t−Mδ+x)] =u(t,x) =EG[ϕ(Mt+x)].

Without loss of generality, by the definition of G-Brownian motion, see[23], we only need to prove the case form=2:

EG[ϕ(Mt1,Mt2−Mt1)]

= EG[EG[ϕ(Mt1,Mt2−Mt1)|Ft1]] = E_G[E_G[ϕ(x,M_t

2−Mt1)|Ft1]x=Mt1] = E_G[E_G[ϕ(x,M_t2−t1)]_x=M

t1].

Based on Step 1_∼Step 4, M has the same finite distribution with G-Brownian motionB, we com-plete the proof of the Theorem 5.1.

6

D

ISCUSSION

In this paper, we investigate the properties of capacity defined by expectation, and prove that G-expectation is not filtration consistent. Meanwhile by path-wise analysis, we prove that symmetric G-martingales are martingales under eachP_v. Based on these arguments, we obtain the martingale characterization theorem for a G-Brownian motion. The application of this framework in finance can be found in[9]. Our martingale characterization theorem of G-Brownian motion includes Lèvy’s martingale characterization theorem of Brownian motion. In the proof of classical Lèvy’s martingale characterization theorem, the characterization function of normal distribution plays an important

Our proof of martingale characterization of G-Brownian motion gives a totally different method to prove Lèvy’s martingale characterization of Brownian motion. In our proof the compactness ofΛis essential. Based on our result one can investigate some elementary problems such as the martingale representation with respect to G-Brownian motion in G-framework.

7

A

PPENDIX

Lemma 7.1. If M∈L1_G(_Ft),ϕ(x)is Lipschitz continuous, thenϕ(M(t))_∈L1_G(_Ft).

Proof: Without loss of generality, supposeϕ(x)_≥0. We know that there exists a sequenceϕn(x)∈ l i p(R) such thatϕn(x)−→ϕ(x)monotonically. By the Lipschitz continuous ofϕ(x), and the fact thatE_G[_|ϕ(M_t)_|2_{]<∞, we have}

E_G[_|ϕ(M_t)₋ϕ_n(M_t)_|]_≤E_G[_|ϕ(M_t)_|I_{|Mt|≥n}]

≤ E_G1/2[_|ϕ(M_t)_|2]E_G1/2[I_{|M

t|≥n}]≤E

1/2

G [|ϕ(Mt)|2]

E_G1/2[M_t2]

n −→0.

Proposition 7.2. For any v ∈Λ′, set Iδ = E[sup_|s−t|≤δ| Z t

s

vτd Bτ|], thenlimδ→0Iδ =0, where E

refers to the expectation under Weiner measure, and B is the canonical process which is the Brownian motion under Weiner measure. (see the definition ofΛ′in §3 )

Proof:The proof is trivial, and hence omitted.

Acknowledgements

We would like to thank the anonymous referee for carefully reading the

manuscript and offering useful suggestions and comments which improved the clarity of the paper. We would like to thank Professor Bingyi Jing from Hong Kong University of Science and Technology for his constructive advice. Some results of this work were carried out while Dr. Xu was visiting Shandong University and Southern California University. She would like to thank Professor Shige Peng, Professor Jin Ma and Professor Jianfeng Zhang for their hospitality, suggestions and helpful discussions which greatly improved the presentation.

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