Characterizations of processes with stationary and independent increments under <span class="mathjax-formula">$G$</span>-expectation

www.imstat.org/aihp 2013, Vol. 49, No. 1, 252–269

DOI:10.1214/12-AIHP492

© Association des Publications de l’Institut Henri Poincaré, 2013

Characterizations of processes with stationary and independent increments under G-expectation ¹

Yongsheng Song

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China. E-mail:yssong@amss.ac.cn Received 26 October 2010; revised 8 April 2012; accepted 17 April 2012

Abstract. Our purpose is to investigate properties for processes with stationary and independent increments underG-expectation.

As applications, we prove the martingale characterization ofG-Brownian motion and present a pathwise decomposition theorem for generalizedG-Brownian motion.

Résumé. Notre but est d’étudier des propriétés de processus à accroissements stationnaires et indépendants sous uneG-espérance.

Comme application, nous démontrons la caractérisation de la martingale deG-mouvement Brownien et fournissons un théorème de décomposition trajectorielle pour leG-mouvement Brownien généralisé.

MSC:60G10; 60G17; 60G48; 60G51

Keywords:Stationary increments; Independent increments; Martingale characterization; Decomposition theorem;G-Brownian motion;

G-expectation

1. Introduction

Recently, motivated by the modelling of dynamic risk measures, Shige Peng ([3–5]) introduced the notion of aG- expectation space. It is a generalization of probability spaces (with their associated linear expectation) to spaces endowed with a nonlinear expectation. As the counterpart of Wiener space in the linear case, the notion ofG-Brownian motion was introduced under the nonlinearG-expectation.

Recall that if {A_t}is a continuous process over a probability space(Ω,F, P )with stationary, independent in- crements and finite variation, then there exists some constantcsuch thatA_t =ct. However, it is not the case in the G-expectation space (Ω_T, L¹_G(Ω_T),E). A counterexample isˆ {Bt}, the quadratic variation process for the coor- dinate process{B_t}, which is aG-Brownian motion. We know that {Bt}is a continuous, increasing process with stationary and independent increments, but it is not deterministic.

The process{Bt}is very important in the theory ofG-expectation, which shows, in many aspects, the difference between probability spaces andG-expectation spaces. For example, we know that for a probability space continuous local martingales with finite variation are trivial processes. However, [4] proved that in aG-expectation space all pro- cesses in form of_t

0η_sdBs−_t

02G(ηs)ds,η∈M_G¹(0, T )(see Section2for the definitions of the functionG(·)and the spaceM_G¹(0, T )), are nontrivialG-martingales with finite variation (in fact, they are even nonincreasing) and con- tinuous paths. [4] also conjectured that anyG-martingale with finite variation should have such representation. Up to

1Supported by NCMIS; Youth Grant of National Science Foundation (No. 11101406/A011002); the National Basic Research Program of China (973 Program) (No. 2007CB814902); Key Lab of Random Complex Structures and Data Science, Chinese Academy of Sciences (Grant No. 2008DP173182).

now, some properties of the process{Bt}remain unknown. For example, we know that, ifG(x)=¹₂sup_σ≤σ≤σσ²x generates theG-expectation, we haveσ²(t−s)≤ Bt− Bs≤σ (t−s)for alls < t, but we do not know whether {_ds^dBs}belongs toM_G¹(0, T ). This is a very important property since{_ds^dBs} ∈M_G¹(0, T )would imply that the representation mentioned above ofG-martingales with finite variation is not unique.

For the case of a probability space, a continuous local martingale{M_t}is a standard Brownian motion if and only if the quadratic variation process Mt =t. However, it’s not the case for G-Brownian motion since its quadratic variation process is only an increasing process with stationary and independent increments. How can we give a char- acterization forG-Brownian motion?

In this article, we shall prove that if At=t

0hsds (respectivelyAt =t

0hsdBs) is a process with stationary, independent increments and h∈M_G¹(0, T )(respectivelyh∈M_G^β,+(0, T ), for someβ >1), then there exists some constantcsuch thath≡c. As applications, we prove the following conclusions (Question 1 and 3 are put forward by Prof. Shige Peng in private communications):

1. {_ds^dBs}∈/M_G¹(0, T ).

2. (Martingale characterization)

A symmetricG-martingale{M_t}is aG-Brownian motion if and only if its quadratic variation process{Mt} has stationary and independent increments;

A symmetricG-martingale{M_t}is aG-Brownian motion if and only if its quadratic variation processMt= cBt for somec≥0.

The sufficiency of the second assertion is trivial, but not the necessity.

3. Let{X_t}be a generalizedG-Brownian motion with zero mean, then we have the following decomposition:

Xt=Mt+Lt,

where{M_t}is a (symmetric) G-Brownian motion, and {L_t}is a nonpositive, nonincreasingG-martingale with stationary and independent increments.

This article is organized as follows: In Section2we recall some basic notions and results ofG-expectation and the related space of random variables. In Section3we characterize processes with stationary and independent increments.

In Section4, as application, we prove the martingale characterization ofG-Brownian motion and present a decompo- sition theorem for generalizedG-Brownian motion. In Section5we present some properties forG-martingales with finite variation.

2. Preliminary

We recall some basic notions and results ofG-expectation and the related space of random variables. More details of this section can be found in [3–8].

Definition 2.1. LetΩbe a given set and letHbe a vector lattice of real valued functions defined onΩwithc∈Hfor all constantsc.His considered as the space of “random variables.” A sublinear expectationEˆ onHis a functional Eˆ:H→Rsatisfying the following properties:For allX, Y∈H,we have

(a) Monotonicity:IfX≥Y thenE(X)ˆ ≥ ˆE(Y ).

(b) Constant preserving:E(c)ˆ =c.

(c) Sub-additivity:E(X)ˆ − ˆE(Y )≤ ˆE(X−Y ).

(d) Positive homogeneity:E(λX)ˆ =λE(X),ˆ λ≥0.

(Ω,H,E)ˆ is called a sublinear expectation space.

Definition 2.2. Let X1and X2 be twon-dimensional random vectors defined respectively in sublinear expectation spaces(Ω₁,H1,Eˆ₁)and(Ω₂,H2,Eˆ₂).They are called identically distributed,denoted byX₁∼X₂,ifEˆ₁[ϕ(X₁)] = Eˆ2[ϕ(X2)],for allϕ∈Cl,Lip(Rⁿ),whereCl,Lip(Rⁿ)is the space of real continuous functions defined onRⁿsuch that

ϕ(x)−ϕ(y)≤C

1+ |x|^k+ |y|^k

|x−y|, for allx, y∈Rⁿ,

wherekandCdepend only onϕ.

Definition 2.3. In a sublinear expectation space(Ω,H,E)ˆ a random vectorY =(Y1, . . . , Yn),Yi∈H,is said to be independent of another random vectorX=(X₁, . . . , X_m),X_i∈H,underE(ˆ ·),denoted byY ⊥X,if for every test functionϕ∈C_b,Lip(R^m×Rⁿ)we haveEˆ[ϕ(X, Y )] = ˆE[ ˆE[ϕ(x, Y )]x=X].

Definition 2.4 (G-normal distribution). Ad-dimensional random vectorX=(X1, . . . , X_d)in a sublinear expecta- tion space(Ω,H,E)ˆ is calledG-normal distributed if for everya, b∈R₊we have

aX+bXˆ ∼

a²+b²X,

whereXˆ is an independent copy ofX.Here the letterGdenotes the function G(A):=1

2Eˆ

(AX, X)

:S_d→R,

whereS_ddenotes the collection ofd×dsymmetric matrices.

The functionG(·):S_d→Ris a monotonic, sublinear mapping onS_dandG(A)=¹₂Eˆ[(AX, X)] ≤¹₂|A| ˆE[|X|²] =:

1

2|A| ¯σ²implies that there exists a bounded, convex and closed subsetΓ ⊂S_d⁺such that G(A)=1

2 sup

γ∈Γ

Tr(γ A). (2.1)

If there exists someβ >0 such thatG(A)−G(B)≥βTr(A−B)for anyA≥B, we call theG-normal distribution nondegenerate. This is the case we consider throughout this article.

Definition 2.5. (i)LetΩ_T =C₀([0, T];R^d)be endowed with the supremum norm and{B_t}be the coordinate process.

Set H_T⁰ := {ϕ(B_t1, . . . , B_tn)|n≥1, t1, . . . , t_n∈ [0, T], ϕ∈Cl,Lip(R^d×n)}. G-expectation is a sublinear expectation defined by

Eˆ[X] = ˜E ϕ(√

t₁−t₀ξ₁, . . . ,

t_m−t_m−1ξ_m) ,

for allX=ϕ(B_t1−B_t0, B_t2−B_t1, . . . , B_tm−B_tm−1),whereξ₁, . . . , ξ_n are identically distributedd-dimensionalG- normally distributed random vectors in a sublinear expectation space(Ω,˜ H˜,E)˜ such thatξ_i+1 is independent of (ξ1, . . . , ξi)for everyi=1, . . . , m−1.(ΩT,H⁰_T,E)ˆ is called aG-expectation space.

(ii)Let us define the conditionalG-expectationEˆ_tofξ∈H⁰_T knowingH⁰t,fort∈ [0, T].Without loss of generality we can assume thatξ has the representationξ =ϕ(B_t1 −B_t0, B_t2 −B_t1, . . . , B_tm−B_tm−1) witht =t_i,for some 1≤i≤m,and we put

Eˆ_ti

ϕ(B_t1−B_t0, B_t2−B_t1, . . . , B_tm−B_tm−1)

= ˜ϕ(B_t1−B_t0, B_t2−B_t1, . . . , B_ti−B_ti−1), where

˜

ϕ(x₁, . . . , x_i)= ˆE

ϕ(x₁, . . . , x_i, B_ti+1−B_ti, . . . , B_tm−B_tm−1) .

Defineξp,G= [ ˆE(|ξ|^p)]^1/pforξ∈H⁰_T andp≥1. Then for allt∈ [0, T],Eˆt(·)is a continuous mapping onH⁰_T with respect to the norm · 1,Gand therefore can be extended continuously to the completionL¹_G(Ω_T)ofH⁰_T under the norm · 1,G.

LetL_ip(Ω_T):= {ϕ(B_t1, . . . , B_tn)|n≥1, t1, . . . , t_n∈ [0, T], ϕ∈C_b,Lip(R^d×n)}, whereC_b,Lip(R^d×n)denotes the set of bounded Lipschitz functions onR^d×n. [1] proved that the completions of Cb(ΩT),H⁰_T andLip(ΩT)under · p,Gare the same; we denote them byL^p_G(ΩT).

Definition 2.6. (i)We say that{X_t}on(Ω_T, L¹_G(Ω_T),E)ˆ is a process with independent increments if for any0< t <

T ands₀≤ · · · ≤s_m≤t≤t₀≤ · · · ≤t_n≤T,

(X_t1−X_t0, . . . , X_tn−X_tn−1)⊥(X_s1−X_s0, . . . , X_sm−X_sm−1).

(ii)We say that{Xt}on(ΩT, L¹_G(ΩT),E)ˆ withXt∈L¹_G(Ωt)for everyt∈ [0, T]is a process with independent incrementsw.r.t. the filtrationif for any0< s < T ands0≤ · · · ≤sm≤s≤t0≤ · · · ≤tn≤T,

(X_t1−X_t0, . . . , X_tn−X_tn−1)⊥(B_s1−B_s0, . . . , B_sm−B_sm−1).

Remark 2.7. (i) Let ξ ∈ L¹_G(Ω_T). If there exists s ∈ [0, T] such that for any s₀≤ · · · ≤s_m ≤s, ξ ⊥(B_s1 − B_s0, . . . , B_sm −B_sm−1),then we have Eˆ_s(ξ )= ˆE(ξ ).In fact,there is no loss of generality,we assume E(ξ )ˆ =1 andC≥ξ≥εfor someC, ε >0.Setη= ˆEs(ξ ).For anyn∈N,we have

Eˆ ηⁿ⁺¹

= ˆE ηⁿξ

.

Sinceξ⊥ηⁿ,we have Eˆ

ηⁿ⁺¹

= ˆE ηⁿ

= · · · = ˆE(η)=1.

By this,we have η≤1, q.s.

On the other hand,we have Eˆ

(η−1)²

= ˆE

η(η−2)

+1= ˆE

η(ξ−2) +1.

Sinceξ−2⊥η,we have Eˆ

(1−η)²

= ˆE(1−η).

By Theorem2.12below,there existsP ∈Psuch that EP

(1−η)²

= ˆE

(1−η)² .

Noting that

E_P(1−η)≤ ˆE(1−η)= ˆE

(1−η)²

=E_P

(1−η)²

≤E_P(1−η), we have

E_P

(1−η)²

=E_P(1−η).

By this,we have η²=η, P-a.s.

Sinceη≥ε,we haveη=1,P-a.s.So we have Eˆ

(1−η)²

=E_P

(1−η)²

=0.

(ii)Let{Xt}on(ΩT, L¹_G(ΩT),E)ˆ be a process with stationary and independent increments and letc= ˆE(XT)/T. IfE(Xˆ _t)→0ast↓0,then for any0≤s < t≤T,we haveE(Xˆ _t−X_s)=c(t−s).

Definition 2.8. Let{X_t}be ad-dimensional process defined on(Ω_T, L¹_G(Ω_T),E)ˆ such that:

(i) X₀=0;

(ii) {X_t}is a process with stationary and independent increments w.r.t.the filtration;

(iii) limt→0Eˆ[|Xt|³]t⁻¹=0.

Then{X_t}is called a generalizedG-Brownian motion.

If in additionE(Xˆ _t)= ˆE(−X_t)=0for allt∈ [0, T],{X_t}is called a(symmetric)G-Brownian motion.

Remark 2.9. (i)Clearly,the coordinate process{B_t}is a(symmetric)G-Brownian motion and its quadratic variation process{Bt}is a process with stationary and independent increments(w.r.t.the filtration).

(ii) [4]gave a characterization for the generalizedG-Brownian motion:Let{X_t}be a generalizedG-Brownian motion.Then

X_t+s−X_t∼√

sξ+sη fort, s≥0, (2.2)

where(ξ, η)isG-distributed(see,e.g., [6]for the definition ofG-distributed random vectors).In fact,this character- ization presented a decomposition of generalizedG-Brownian motion in the sense of distribution.In this article,we shall give a pathwise decomposition for the generalizedG-Brownian motion.

LetH_G⁰(0, T )be the collection of processes of the following form: for a given partition{t₀, . . . , t_N} =π_T of[0, T], N≥1,

ηt(ω)=

N−1

j=0

ξj(ω)1_]t_j,t_j+1](t),

whereξ_i∈L_ip(Ω_ti),i=0,1,2, . . . , N−1. For everyη∈H_G⁰(0, T ), letη_H^p

G= { ˆE(_T

0 |η_s|²ds)^p/2}^1/p,η_M^p

G=

{ ˆE(_T

0 |η_s|^pds)}^1/pand denote byH_G^p(0, T ),M_G^p(0, T )the completions ofH_G⁰(0, T )under the norms·_H^p

G,·_M^p respectively. G

Definition 2.10. For everyη∈H_G⁰(0, T )with the form ηt(ω)=

N−1

j=0

ξj(ω)1_]t_j,t_j+1](t),

we define

I (η)= _T

0

η(s)dBs:=

N−1

j=0

ξ_j(B_tj+1−B_tj).

By B–D–G inequality (see Proposition 4.3 in [10] for this inequality under G-expectation), the mapping I:H_G⁰(0, T )→L^p_G(Ω_T)is continuous under · _H^p

G and thus can be continuously extended toH_G^p(0, T ).

Definition 2.11. (i)A process{M_t}with values inL¹_G(Ω_T)is called aG-martingale ifEˆ_s(M_t)=M_s for anys≤t.

If{M_t}and{−M_t}are bothG-martingales,we call{M_t}a symmetricG-martingale.

(ii)A random variableξ∈L¹_G(Ω_T)is called symmetric ifE(ξ )ˆ + ˆE(−ξ )=0.

AG-martingale{Mt}is symmetric if and only ifMT is symmetric.

Theorem 2.12 ([1,2]). There exists a tight subsetP⊂M1(Ω_T)such that E(ξ )ˆ =max

P∈PE_P(ξ ) for allξ∈H⁰_T. Pis called a set that representsE.ˆ

Remark 2.13. (i)Let(Ω⁰,F⁰, P⁰)be a probability space and{W_t}be ad-dimensional Brownian motion underP⁰. LetF⁰= {Ft⁰}be the augmented filtration generated byW. [1]proved that

PM:=

P_h|P_h=P⁰◦X⁻¹, X_t= t

0

h_sdWs, h∈L²_F0

[0, T];Γ^1/2

is a set that represents E,ˆ where Γ^1/2:= {γ^1/2|γ ∈Γ}and Γ is the set in the representation of G(·)in the for- mula(2.1).

(ii)For the1-dimensional case,i.e.,Ω_T =C₀([0, T], R), L²_F0:=L²_F0

[0, T];Γ^1/2

=

h|his adapted w.r.t.F⁰andσ≤h_s≤σ ,

whereσ²= ˆE(B₁²)andσ²= − ˆE(−B₁²).

G(a)=1/2Eˆ aB₁²

=1/2

σ²a⁺−σ²a⁻

fora∈R.

(iii)Setc(A)=sup_P∈PMP (A),forA∈B(Ω_T).We sayA∈B(Ω_T)is a polar set ifc(A)=0.If an event happens except on a polar set,we say the event happens q.s.

3. Characterization of processes with stationary and independent increments

In what follows, we only consider the G-expectation space(Ω_T, L¹_G(Ω_T),E)ˆ withΩ_T =C₀([0, T], R)andσ²= E(Bˆ ₁²) >− ˆE(−B₁²)=σ²>0.

Lemma 3.1. Forζ∈M_G¹(0, T )andε >0,let ζ_t^ε=1

ε _t

(t−ε)⁺

ζ_sds and

ζ_t^ε,0=

k_ε−1

k=1

1 ε

_kε

(k−1)ε

ζ_sds1_]kε,(k+1)ε](t), wheret∈ [0, T],kεε≤T < (kε+1)ε.Then asε→0

ζ^ε−ζ

M_G¹(0,T )→0 and ζ^ε,0−ζ

M_G¹(0,T )→0.

Proof. The proofs of the two cases are similar. Here we only prove the second case. Our proof starts with the obser- vation that for anyζ, ζ∈M_G¹(0, T )

ζ^ε,0−ζ^ε,0

M_G¹(0,T )≤ζ−ζ

M_G¹(0,T ). (3.1)

By the definition of the spaceM_G¹(0, T ), we know that for everyζ∈M_G¹(0, T ), there exists a sequence of processes {ζⁿ}with

ζ_tⁿ=

mn−1

k=0

ξ_tⁿn

k1_]t_kⁿ,t_k+1ⁿ ](t)

andξ_tⁿn

k ∈L_ip(Ω_tⁿ

k)such that ζ−ζⁿ

M_G¹(0,T )→0 asn→ ∞. (3.2)

It is easily seen that for everyn ζ^n;ε,0−ζⁿ

M_G¹(0,T )→0 asε→0. (3.3)

Thus we get ζ^ε,0−ζ

M_G¹(0,T )

≤ζ^ε,0−ζ^n;ε,0

M_G¹(0,T )+ζⁿ−ζ^n;ε,0

M_G¹(0,T )+ζⁿ−ζ

M_G¹(0,T )

≤2ζⁿ−ζ

M_G¹(0,T )+ζⁿ−ζ^n;ε,0

M_G¹(0,T ).

The second inequality follows from (3.1). Combining (3.2) and (3.3), first lettingε→0, then lettingn→ ∞, we have ζ^ε,0−ζ

M_G¹(0,T )→0 asε→0.

Theorem 3.2. LetA_t=_t

0h_sdswithh∈M_G¹(0, T )be a process with stationary and independent increments(w.r.t.

the filtration).Then we haveh≡cfor some constantc.

Proof. Letc:= ˆE(A_T)/T ≥ − ˆE(−A_T)/T =:c. Forn∈N, setε=T /(2n), and defineh^{T /(2n),0}as in Lemma3.1.

Then we have h−h^{T /(2n),0}

M_G¹(0,T )

= ˆE _2n−1

k=0

_{(k+1)T /(2n)}

kT /(2n)

hs−h^{T /(2n),0}_s ds

≥ ˆE _n−1

k=1

(2k+1)T /(2n) 2kT /(2n)

hs−h^{T /(2n),0}_s ds

= ˆE _n−1

k=1

(2k+1)T /(2n) 2kT /(2n)

hsds−

2kT /(2n) (2k−1)T /(2n)

hsds

= ˆE

n−1

k=1

(A_{(2k+1)T /2n}−A2kT /2n)−(A2kT /2n−A_{(2k−1)T /2n}) .

Consequently, from the condition of independence of the increments and their stationarity, we have h−h^{T /(2n),0}

M_G¹(0,T )

≥

n−1

k=1

Eˆ

(A_{(2k+1)T /2n}−A_{2kT /2n})−(A_{2kT /2n}−A_{(2k−1)T /2n})

=

n−1

k=1

(c−c)T /(2n)

=(c−c)(n−1)T /(2n).

So by Lemma3.1, lettingn→ ∞, we havec=c. Furthermore, we note thatM_t:=A_t−ctis aG-martingale. In fact, fort > s, we see

Eˆ_s(M_t)

= ˆE_s(M_t−M_s)+M_s

= ˆE(M_t−M_s)+M_s

=Ms.

The second equality is due to the independence of increments ofMw.r.t. the filtration.

So {M_t} is a symmetric G-martingale with finite variation, from which we conclude that M_t ≡0, hence that

A_t=ct.

Corollary 3.3. Assumeσ > σ >0.Then we have that{_ds^dBs}∈/M_G¹(0, T ).

Proof. The proof is straightforward from Theorem3.2.

Corollary 3.4. There is no symmetricG-martingale{Mt}which is a standard Brownian motion underG-expectation (i.e.Mt=t).

Proof. Let{M_t}be a symmetricG-martingale. If{M_t}is also a standard Brownian motion, by Theorem 4.8 in [10]

or Corollary 5.2 in [11], there exists{h_s} ∈M_G²(0, T )such that M_t=

_t

0

h_sdBs

and _t

0

h²_sdBs=t.

Thus we have _ds^dBs=h⁻_s²∈M_G¹(0, T ), which contradicts the conclusion of Corollary3.3.

Proposition 3.5. LetAt =_t

0hsdswithh∈M_G¹(0, T )be a process with independent increments.ThenAt is sym- metric for everyt∈ [0, T].

Proof. By arguments similar to those in the proof of Theorem3.2, we have h−h^{T /(2n),0}

M_G¹(0,T )

≥ ˆE

n−1

k=0

(A_{(2k+1)T /2n}−A_{2kT /2n})−(A_{2kT /2n}−A_(2k−1)+T /2n)

=

n−1

k=0

E(Aˆ _{(2k+1)T /2n}−A_{2kT /2n})+ ˆE

−(A_{2kT /2n}−A_(2k−1)+T /2n) .

The right side of the first inequality is only the sum of the odd terms. Summing up the even terms only, we have h−h^{T /(2n),0}

M_G¹(0,T )

≥

n−1

k=0

E(Aˆ (2k+2)T /2n−A(2k+1)T /2n)+ ˆE

−(A(2k+1)T /2n−A2kT /2n) .

Combining the above inequalities, we have 2h−h^{T /(2n),0}

M_G¹(0,T )

≥

2n−1

k=0

Eˆ

A_{(k+1)T /2n}−A_{kT /2n} + ˆE

−(A_{(k+1)T /2n}−A_{kT /2n})

≥ ˆE

2n−1

k=0

A_{(k+1)T /2n}−A_{kT /2n} + ˆE

2n−1

k=0

−(A_{(k+1)T /2n}−A_{kT /2n})

= ˆE(A_T)+ ˆE(−A_T).

Thus by Lemma3.1, lettingn→ ∞, we haveE(Aˆ T)+ ˆE(−AT)=0, which means thatAT is symmetric.

Forn∈N, defineδ_n(s)in the following way:

δ_n(s)=

n−1

i=0

(−1)ⁱ1_]iT

n,⁽ⁱ⁺_n^1)T](s) for alls∈ [0, T]. In [12] we proved that limn→∞E(ˆ _T

0 δ_n(s)h_sds)=0 forh∈M_G¹(0, T ).

LetFt=σ{B_s|s≤t}andF= {Ft}t∈[0,T].

In the following, we shall use some notations introduced in Remark2.13.

For every P ∈PM and t ∈ [0, T], set At,P := {Q∈PM|Q_|Ft =P_|Ft}. Proposition 3.4 in [9] gave the fol- lowing result: For t ∈ [0, T], assume ξ ∈L¹_G(Ω_T) and η∈L¹_G(Ω_t). Then η= ˆE_t(ξ ) if and only if for every P ∈PM

η=ess sup

Q∈At,P

PEQ(ξ|Ft), P-a.s.,

where ess sup^P denotes the essential supremum underP. Theorem 3.6. Let A_t =_t

0h_sdBs be a process with stationary,independent increments(w.r.t.the filtration)and h∈M_G^1,+(0, T ).IfA_T ∈L^β_G(Ω_T)for someβ >1,we haveA_t=cBt for some constantc≥0.

Proof. For the readability, we divide the proof into several steps:

Step1. SetKt:=t

0hsds. We claim thatKT is symmetric.

Step1.1. Letμ= ˆE(A_T)/T andμ= − ˆE(−A_T)/T. First, we shall prove that ^μ

σ² =_σ^μ2. Actually, for any 0≤s < t≤T, we have

Eˆ_{s t}

s

h_rdr

= ˆE_{s t}

s

θ_r⁻¹dAr

≥ 1 σ²Eˆ_s

_t

s

dAr

= μ

σ²(t−s) q.s.,

where the inequality holds due toθ_s := ^d_ds^Bs ≤σ², q.s. Noting that μt −A_t is nonincreasing by Lemma 4.3 in Section4since it is aG-martingale with finite variation, we have, for everyη∈L²

F⁰,Pη-a.s., Eˆ_s

t s

h_rdr

=ess sup

Q∈At,Pη

PηE_Q t

s

h_rdrFs

=ess sup

Q∈At,Pη

PηE_Q t

s

θ_r⁻¹dArFs

≥μess sup

Q∈At,Pη

PηEQ

t s

θ_r⁻¹drFs

= μ σ²(t−s).

SoEˆ_s(_t

s h_rdr)≥max{_σ^μ2, ^μ

σ²}(t−s)=:λ(t−s), q.s.

On the other hand, Eˆ_s

− _t

s

h_rdr

= ˆE_{s t}

s

−θ_r⁻¹dAr

≥ 1 σ²Eˆ_s

− _t

s

dAr

= −μ

σ²(t−s), q.s.

and for everyη∈L²

F⁰,P_η-a.s., ˆ

E_s

− _t

s

h_rdr

=ess sup

Q∈At,Pη

PηE_Q

− _t

s

h_rdrFs

=ess sup

Q∈At,Pη

PηE_Q

− t

s

θ_r⁻¹dArFs

≥μess sup

Q∈At,Pη

PηE_Q

− _t

s

θ_r⁻¹drFs

= −μ σ²(t−s)

sinceAt−μt is nonincreasing. So Eˆs

− _t

s

hrdr

≥ −min μ

σ², μ σ²

(t−s)=: −λ(t−s), q.s.

Noting that Eˆ

_T

0

δ_2n(s)h_sds

= ˆE

(2n−1)T /(2n) 0

δ_2n(s)h_sds+ ˆE_{(2n−1)T /(2n)}

− _T

(2n−1)T /(2n)

h_sds

≥(−λ)T 2n+ ˆE

(2n−2)T /(2n) 0

δ_2n(s)h_sds+ ˆE_{(2n−2)T /(2n)}

(2n−1)T /(2n) (2n−2)T /(2n)

h_sds

≥λ−λ 2n T + ˆE

_{(2n−2)T /(2n)}

0

δ_2n(s)h_sds

,

we have Eˆ

T 0

δ2n(s)h_sds

≥λ−λ 2 T .

So

0= lim

n→∞Eˆ _T

0

δ_2n(s)h_sds

≥ λ−λ 2 T and ^μ

σ² =_σ^μ2 =:λ.

Step1.2. For everyη∈L²

F⁰,E_Pη(K_T)=λT, which implies thatK_T is symmetric.

Step1.2.1. We now introduce some notations: For 0≤s < t≤T andη∈L²

F⁰, setη=σ,η=σ,η^∗=

σ²+σ²

2 on

]s, t]andη=η=η^∗=ηon]s, t]^c. Forn∈N, setη_rⁿ=n−1

i=0(σ1_]t_2i,t_2i+1](r)+σ1_]t_2i+1,t_2i+2](r))on]s, t]andηⁿ=η on]s, t]^c, wheret_j=s+_2n^j (t−s),j=0, . . . ,2n.

Step1.2.2.E_Pηn(_t

s(h_r−λ)dr|Fs)→0,P_η-a.s., asn→ ∞.

Actually, we have,P_η-a.s., μ(t−s)= ˆE_s

t s

h_rdBr

≥E_Pη t

s

h_rdBrFs

=σ²E_Pη t

s

h_rdrFs

.

So E_Pη

_t

s

h_rdrFs

≤λ(t−s), P_η-a.s. (3.4)

By similar arguments we have that E_Pη

_t

s

h_rdrFs

≥λ(t−s), P_η-a.s. (3.5)

Let’s compute the following conditional expectations:

E_{Pηn t}

s

(h_r−λ)δ_2n(r)drFs

=E_P^Fs

ηn

_n−1

i=0

E^F_P^t2i

ηn

_t2i+1

t2i

(h_r−λ)dr+E_P^Ft2i+1

ηn

_t2i+2

t_2i+1

(λ−h_r)dr

=:E_P^Fs

ηn

_n−1

i=0

(A_i+B_i)

,

whereδ_2n(r)=_n−1

i=0(1_]t_2i,t_2i+1](r)−1_]t_2i+1,t_2i+2](r)),t_j=s+_2n^j (t−s),j=0, . . . ,2n;

E_{Pηn t}

s

(h_r−λ)drFs

=E_P^Fs

ηn

_n−1

i=0

(A_i−B_i)

.

By (3.4) and (3.5) (noting thatηands, tare all arbitrary), we conclude thatA_i, B_i≥0,P_ηn-a.s. So EP_ηn

t s

(hr−λ)drFs

≤EP_ηn

t s

(hr−λ)δ2n(r)drFs

, Pη-a.s.

Noting that E_Pηn

t s

(h_r−λ)δ2n(r)drFs

≤ ˆE_s t

s

(h_r−λ)δ2n(r)dr

, P_η-a.s.