A Note on the Exponential G-Martingale

Abstract

We get the exponential G-martingale theorem with the Kazamaki condition and tell a distinct difference between the Kazamaki’s and Novikov’s criteria with an example.

1. Introduction and Main Result

Motivated by various types of uncertainty and financial problems, Peng [1] has introduced a new notion of nonlinear expectation, the so-called 𝐺-expectation (see also Peng [2]), which is associated with the following nonlinear heat equation: 𝜕𝑢()(),()∈[)𝑢()(𝑥),𝜕𝑡𝑡,𝑥=𝐺Δ𝑢𝑡,𝑥0,+∞×ℝ,0,𝑥=𝜑 (1) where Δ is Laplacian and the sublinear function 𝐺 is defined by 𝐺(𝛼)=12𝜎²𝛼⁺−𝜎²𝛼⁻,𝛼∈ℝ, (2) with two given constants 0<𝜎<𝜎. Together with the notion of 𝐺-expectations, Peng also introduced the related 𝐺-normal distribution, the 𝐺-Brownian motion, and related stochastic calculus under 𝐺-expectation, and moreover an Itô's formula for the 𝐺-Brownian motion was established. 𝐺-Brownian motion has a very rich and interesting new structure which nontrivially generalizes the classical one. Briefly speaking, a 𝐺-Brownian motion 𝐵 is a continuous process with independent stationary increments 𝐵_𝑡+𝑠−𝐵_𝑡 being 𝐺-normally distributed under a given sublinear expectation  𝔼. A very interesting new phenomenon of 𝐺-Brownian motion 𝐵 is that its quadratic process ⟨𝐵⟩ is a continuous process with independent and stationary increments, but not a deterministic process.

Recently, Xu et al. [3] got an exponential martingale theorem under 𝐺-framework with an assumption of Novikov’s type. In this note, we will introduce the sublinear version of the classical Kazamaki condition. The main objective is to explain and prove the following theorem.

Theorem 1. If there exists an 𝜀₀>0 such that 𝔼1exp2+𝜀₀^𝑇₀𝐻(𝑠,𝜔)𝑑𝐵_𝑠<∞, (3) then ℰ𝐵_𝑡∶=exp^𝑡₀𝐻(𝑠,𝜔)𝑑𝐵_𝑠−12^𝑡₀𝐻²(𝑠,𝜔)𝑑⟨𝐵⟩_𝑠 (4) is a symmetric martingale under 𝔼.

Under the classical case, the result is called Kazamaki's condition, and it can be recalled as follows, for any classical continuous martingale 𝑀, if 1𝐸exp2𝑀_∞<∞, (5) then the martingale exp𝑀_𝑡−12⟨𝑀⟩_𝑡,0≤𝑡≤∞, (6) is uniformly integrable. Clearly, the Kazamaki principle is weaker than the Novikov condition.

This note is organized as follows. In Section 2, we present some standard concepts and notations about 𝐺-Brownian motion and 𝐺-Expectation. In Section 3 we prove the above theorem and discuss some examples.

2. Preliminaries

In this section, we recall some concepts under the 𝐺-framework which are needed in our analysis. For more details, one can see Peng [1].

Let Ω≠∅ be a given set and let ℋ be a linear space of real valued functions defined on Ω such that 1∈ℋ and |𝑋|∈ℋ for all 𝑋∈ℋ.

Definition 2. A sublinear expectation 𝔼 on ℋ is a functional with the following properties, for all 𝑋,𝑌∈ℋ, one has(i)monotonicity: if 𝑋≥𝑌, then 𝔼[𝑋]≥𝔼[𝑌];(ii)constant preserving:  𝔼[𝑐]=𝑐, for all 𝑐∈ℝ;(iii)subadditivity:𝔼[𝑋]−𝔼[𝑌]≤𝔼[𝑋−𝑌];(iv)positive homogeneity: 𝔼[𝜆𝑋]=𝜆𝔼[𝑋], for all 𝜆≥0.The triple (Ω,ℋ,𝔼) is called a sublinear expectation space, and ℋ is considered as the space of random variables on Ω.

It is important to note that one can suppose that 𝜑𝑋₁,…,𝑋_𝑑∈ℋ (7) if 𝑋_𝑖∈ℋ,𝑖=1,…,𝑑, for all 𝜑∈𝐶_{𝑏,𝐿𝑖𝑝}(ℝ^𝑑), where 𝐶_{𝑏,𝐿𝑖𝑝}(ℝ^𝑑) denotes the space of all bounded and Lipschitz functions on ℝ^𝑑. In a sublinear expectation space (Ω,ℋ,𝔼), a random vector 𝑌=(𝑌₁,…,𝑌_𝑛),𝑌_𝑖∈ℋ, is said to be independent under 𝔼 from another random vector 𝑋=(𝑋₁,…,𝑋_𝑚),𝑋_𝑖∈ℋ, if for each test function 𝜑∈𝐶_{𝑏,𝐿𝑖𝑝}(ℝ^𝑚+𝑛), one has 𝔼𝜑(𝑋,𝑌)=𝔼𝔼𝜑(𝑥,𝑌)_𝑥=𝑋. (8) Two 𝑛-dimensional random vectors 𝑋 and 𝑌 defined, respectively, in the sublinear expectation spaces (Ω₁,ℋ₁,𝔼₁) and (Ω₂,ℋ₂,𝔼₂) are called identically distributed, denoted by 𝑋∼𝑌, if 𝔼₁𝜑(𝑋)=𝔼₂𝜑(𝑌), (9) for all 𝜑∈𝐶_{𝑏,𝐿𝑖𝑝}(ℝ^𝑛).

Let 𝜎 and 𝜎 be two real numbers with 0<𝜎<𝜎. A random variable 𝜉 in a sublinear expectation space (Ω,ℋ,𝔼) is called 𝐺-normal distributed, denoted by 𝜉∼𝑁(0,[𝜎²,𝜎²]), if for each 𝜑∈𝐶_{𝑏,𝐿𝑖𝑝}(ℝ), the function defined by 𝑢(𝑡,𝑥)∶=𝔼𝜑𝑥+√𝑡𝜉,(𝑡,𝑥)∈[0,∞)×ℝ, (10) is the unique viscosity solution of the following nonlinear heat equation: 𝜕𝑢()(),()∈[)𝑢()(𝑥),𝜕𝑡𝑡,𝑥=𝐺Δ𝑢𝑡,𝑥0,+∞×ℝ,0,𝑥=𝜑 (11) where Δ is Laplacian and the sublinear function 𝐺 is defined by 𝐺(𝛼)=12𝜎²𝛼⁺−𝜎²𝛼⁻,𝛼∈ℝ. (12)

Example 3 (Peng [2]). Let 𝜉∼𝑁(0,[𝜎²,𝜎²]). We then have 𝔼𝜑(𝜉=1)√2𝜋𝜎^ℝ𝜑(𝑥)𝑒^{(−1/2𝜎2)𝑥2}𝑑𝑥 (13) for all convex functions 𝜑 and 𝔼𝜓(𝜉=1)√2𝜋𝜎^ℝ𝜑(𝑥)𝑒^{(−1/2𝜎2)𝑥2}𝑑𝑥 (14) for all concave functions 𝜓.

Let now Ω=𝐶₀(ℝ⁺) be the space of all real valued continuous functions on [0,∞) with initial value 0, equipped with the distance 𝜌𝜔¹,𝜔²=^∞_𝑖=12^−𝑖max_{𝑡∈[0,𝑖]}𝜔¹_𝑡−𝜔²_𝑡∧1,𝜔¹,𝜔²∈Ω. (15) We denote by ℬ(Ω) the Borel-algebra on Ω. We also denote, for each 𝑡∈[0,∞), Ω_𝑡=𝜔_⋅∧𝑡,𝜔∈Ω (16) and ℱ_𝑡=ℬ(Ω_𝑡), where 𝑥∧𝑦=min{𝑥,𝑦}. We also denote the following:(i)𝐿⁰(Ω): the space of all ℬ(Ω)-measurable real valued functions on Ω;(ii)𝐿⁰(Ω_𝑡): the space of all ℬ(Ω_𝑡)-measurable real valued functions on Ω_𝑡;(iii)𝐿_𝑏(Ω): the space of all bounded elements in 𝐿⁰(Ω);(iv)𝐿_𝑏(Ω_𝑡): the space of all bounded elements in 𝐿⁰(Ω_𝑡).Let 𝕃^𝑝_𝐺(Ω) be the closure of ℋ with respect to the norm ‖𝑋‖_𝑝=𝔼|𝑋|^𝑝1/𝑝 (17) with 𝑝∈[1,∞). Clearly, the space 𝕃^𝑝_𝐺(Ω) is a Banach space and the space 𝐶_𝑏(Ω) of bounded continuous functions on Ω is a subset of 𝕃¹_𝐺(Ω), and, moreover, for the sublinear expectation space (Ω,𝕃^𝑝_𝐺(Ω),𝔼), there exists a weakly compact family 𝒫 of probability measures on (Ω,ℬ(Ω)) such that 𝔼=sup_𝑃∈𝒫𝐸_𝑃. (18) So we can introduce the Choquet capacity 𝐶 by taking 𝐶(𝐴)=sup_𝑃∈𝒫𝑃(𝐴),𝐴∈ℬ(Ω). (19)

Definition 4. A set 𝐴⊂Ω is called polar if 𝐶(𝐴)=0. A property is said to hold “quasi-surely” (q.s.) if it holds outside a polar set.

By using the above family of probability measures 𝑃, one can characterize the space 𝕃^𝑝_𝐺(Ω) as 𝕃^𝑝_𝐺(Ω)=𝐿⁰(Ω)∋𝑋iscontinuous,q.s.,sup_𝑃∈𝒫𝐸_𝑃|𝑋|^𝑝≡𝐿<∞₀(Ω)∋𝑋iscontinuous,q.s.,lim_𝑛→∞sup_𝑃∈𝒫𝐸_𝑃|𝑋|^𝑝1_{{|𝑋|>𝑛}}=0. (20) The following three results can be consulted in Denis et al. [4] and Hu and Peng [5].

Lemma 5 (Denis et al. [4] and Hu and Peng [5]). Let {𝑋_𝑛,𝑛=1,2,…} be a monotonically decreasing sequence of nonnegative random variances in 𝐶_𝑏(Ω). If 𝑋_𝑛 converges to zero q.s. on Ω, then one has lim_𝑛→0𝔼𝑋_𝑛=0. (21) Moreover, if 𝑋_𝑛↑𝑋 and 𝔼[𝑋] and 𝔼[𝑋_𝑛] are finite for all 𝑛=1,2,…, one then has lim_𝑛→0𝔼𝑋_𝑛=𝔼[𝑋]. (22)

Lemma 6 (Denis et al. [4] and Hu and Peng [5]). Let 1≤𝑝<∞. Consider the sets 𝕃^𝑝_𝐺(Ω) and 𝕃^𝑝=ℒ^𝑝/𝒩, where ℒ^𝑝=𝑋∈𝐿⁰(Ω)∶𝔼|𝑋|^𝑝=sup_𝑃∈𝒫𝐸_𝑃|𝑋|^𝑝<∞,𝒩=𝑋∈𝐿⁰(Ω)∶𝑋=0q.s.. (23) Then,(i)𝕃^𝑝 is a Banach space with respect to the norm ‖⋅‖_𝑝;(ii)𝕃^𝑝_𝐺 is the completion of 𝐶_𝑏(Ω) with respect to the norm ‖⋅‖_𝑝.

Lemma 7 (Denis et al. [4] and Hu and Peng [5]). For a given 𝑝∈(0,+∞], if the sequence 𝕃^𝑝⊃{𝑋_𝑛} converges to 𝑋 in 𝕃^𝑝, then there exists a subsequence {𝑋_𝑛𝑘} such that 𝑋_𝑛𝑘 converges to 𝑋 quasi-surely.

We denote by 𝕃^𝑝_∗(Ω) the completion of 𝐿_𝑏(Ω) with respect to the norm ‖⋅‖_𝑝.

Definition 8 (𝐺-Brownian motion). A process 𝐵={𝐵_𝑡,𝑡≥0}⊂ℋ in a sublinear expectation space (Ω,ℋ,𝔼) is called a 𝐺-Brownian motion if the following properties are satisfied:(i)𝐵₀=0;(ii)for each 𝑡,𝑠≥0, the increment 𝐵_𝑡+𝑠−𝐵_𝑡 is 𝑁(0,[𝜎²𝑠,𝜎²𝑠])-distributed and is independent from (𝐵_𝑡1,…,𝐵_𝑡𝑛), for all 𝑛=0,1,2,… and 0≤𝑡₁≤𝑡₂≤⋯≤𝑡_𝑛≤𝑡.

The 𝐺-Brownian motion 𝐵 has the following properties:(1)for all 𝜉∈𝕃²(Ω_𝑡), one has 𝔼[𝜉(𝐵_𝑇−𝐵_𝑡)]=0 with 0≤𝑡≤𝑇;(2)for all ℬ(Ω_𝑡)-measurable real valued, bounded functions 𝜉, one has 𝔼𝜉²𝐵_𝑇−𝐵_𝑡²≤𝜎²(𝑇−𝑡)𝔼𝜉²,0≤𝑡≤𝑇; (24)(3)for all 𝑡≥0, one has 𝔼[𝐵_𝑡]=𝔼[−𝐵_𝑡]=0;(4)𝑡↦𝐵_𝑡 is Hölder continuous of order 𝛿<1/2, quasi-surely.

In Li and Peng [6], a generalized Itô integral and a generalized Itô formula with respect to the 𝐺-Brownian motion are introduced. For arbitrarily fixed 𝑝≥1 and 𝑇∈ℝ₊, one denotes by 𝑀_𝑏^𝑝,0([0,𝑇]) the set of step processes as follows: 𝜂_𝑡(𝜔)=^𝑁_𝑗=1𝜉_𝑗(𝜔)1_{[𝑡𝑗−1,𝑡𝑗)}(𝑡),𝜉_𝑗∈𝐿_𝑏Ω_𝑡𝑗−1 (25) with 0=𝑡₀<⋯<𝑡_𝑁=𝑇. For the process of the form (25) one defines the related Bochner integral as follows: ^𝑇₀𝜂_𝑡𝑑𝑡=^𝑁_𝑗=1𝜉_𝑗𝑡_𝑗−𝑡_𝑗−1. (26) For every 𝜂∈𝑀_𝑏^𝑝,0([0,𝑇]), one sets 𝔼_𝑇𝜂1∶=𝑇𝔼^𝑇₀𝜂_𝑡𝑑𝑡. (27) Then 𝔼_𝑇 forms a sublinear expectation. Moreover, one denotes by 𝑀^𝑝_∗([0,𝑇]) the completion of 𝑀_𝑏^𝑝,0([0,𝑇]) under the norm 𝜂_𝑀^𝑝_{∗([0,𝑇])}=𝔼^𝑇₀𝜂_𝑠^𝑝𝑑𝑠^1/𝑝. (28)

Definition 9. For every 𝜂∈𝑀_𝑏^𝑝,0([0,𝑇]) of the form (25), one defines the Itô integral of 𝜂 with respect to 𝐺-Brownian motion 𝐵 by 𝐼𝜂∶=^𝑇₀𝜂_𝑠𝑑𝐵_𝑠=^𝑁_𝑗=1𝜉_𝑗𝐵_𝑡𝑗−𝐵_𝑡𝑗−1. (29)

The mapping 𝐼∶𝑀_𝑏^𝑝,0([0,𝑇])→𝕃²_∗(Ω_𝑇) is a linear continuous mapping and thus can be continuously extended to 𝐼∶𝑀²_∗([0,𝑇])→𝕃²_∗(Ω_𝑇), which is called the Itô integral of 𝜂∈𝑀²_∗([0,𝑇]) with respect to 𝐺-Brownian motion 𝐵, and define ^𝑡₀𝜂_𝑠𝑑𝐵_𝑠=^𝑇₀1_{{0≤𝑠≤𝑡}}𝜂_𝑠𝑑𝐵_𝑠 (30) for all 𝜂∈𝑀²_∗([0,𝑇]) and 𝑡∈[0,𝑇]. One has 𝔼^𝑇₀𝜂_𝑠𝑑𝐵_𝑠𝔼=0,_𝑇0𝜂_𝑠𝑑𝐵_𝑠²≤𝜎²𝔼^𝑇₀𝜂²_𝑠𝑑𝑠 (31) for all 𝜂∈𝑀²_∗([0,𝑇]). Moreover, the process {∫^𝑡₀𝜂_𝑠𝑑𝐵_𝑠,𝑡∈[0,𝑇]} is continuous in 𝑡 quasi-surely and ^⋅₀𝜂_𝑠𝑑𝐵_𝑠∈𝑀²_∗([0,𝑇]) (32) for all 𝜂∈𝑀²_∗([0,𝑇]).

Definition 10 (quadratic variation). Let 𝜋^𝑁_𝑡={0=𝑡^𝑁₀<𝑡^𝑁₁<⋯<𝑡^𝑁_𝑁−1=𝑡} be a partition of [0,𝑡] for 𝑡>0, such that 𝜇(𝜋^𝑁_𝑡)∶=max_𝑗{𝑡_𝑗−𝑡_𝑗−1}→0 as 𝑁→∞. The quadratic variation of 𝐺-Brownian motion 𝐵 is defined by ⟨𝐵⟩_𝑡=lim_{𝜇𝜋𝑁𝑡→0}^𝑁−1_𝑘=0𝐵_𝑡^𝑁_𝑘+1−𝐵_𝑡^𝑁_𝑘²=𝐵²_𝑡−2^𝑡₀𝐵_𝑠𝑑𝐵_𝑠 (33) in 𝕃²_𝐺(Ω).

The function 𝑡↦⟨𝐵⟩_𝑡 is continuous and increasing outside a polar set. One can define the integral ^𝑇₀𝜂_𝑡𝑑⟨𝐵⟩_𝑡∶=^𝑁_𝑗=1𝜉_𝑗⟨𝐵⟩_𝑡𝑗−⟨𝐵⟩_𝑡𝑗−1 (34) as a map from 𝑀_𝑏^1,0([0,𝑇]) into 𝕃¹_∗(Ω_𝑇), and the map is linear and continuous, and it can be extended continuously to 𝑀¹_∗([0,𝑇]).

Definition 11. A process (𝑀_𝑡)_𝑡⩾0 is called a 𝐺-martingale if, for each 𝑡∈[0,∞),𝑀_𝑡∈𝕃¹_𝐺(ℱ_𝑡) and, for each 𝑠∈[0,𝑡], one has 𝔼𝑀_𝑡∣ℱ_𝑠=𝑀_𝑠. (35) If both (𝑀_𝑡)_𝑡⩾0 and (−𝑀_𝑡)_𝑡⩾0 are 𝐺-martingales, (𝑀_𝑡)_𝑡⩾0 is called a 𝐺-symmetric martingale.

One can easily prove that the process 𝑀_𝑡∶=𝔼𝑋∣ℱ_𝑡,𝑡≥0, (36) is a 𝐺-martingale for each 𝑋∈𝕃¹_𝐺(ℱ), and, moreover, the process 𝑀_𝑡∶=𝑀₀+^𝑡₀𝜑_𝑢𝑑𝐵_𝑢+^𝑡₀𝜂𝑑⟨𝐵⟩_𝑢−^𝑡₀2𝐺𝜂_𝑢𝑑𝑢,∀𝑡∈[0,𝑇] (37) also is a 𝐺-martingale for all 𝑀₀∈ℝ,𝜑∈𝑀²_𝑏[0,𝑇],𝜂∈𝑀¹_𝑏[0,𝑇].

3. Proof of Theorem 1

Theorem 12. If there exists an 𝜖>0 such that 𝔼1exp2+𝜖₀^𝑇₀𝐻(𝑠,𝜔)𝑑𝐵_𝑠<∞, (38) then ℰ𝐵_𝑡∶=exp^𝑡₀𝐻(𝑠,𝜔)𝑑𝐵_𝑠−12^𝑡₀𝐻²(𝑠,𝜔)𝑑⟨𝐵⟩_𝑠 (39) is a symmetric martingale under 𝔼, and, for all 𝑡∈[0,𝑇],ℰ(𝐵_𝑡)∈𝕃¹_𝐺(ℱ_𝑡).

Proof. Since 𝐵_𝑡 is a mean-square integrable martingale under each 𝑃_𝜈 and 𝐸_𝑃𝜈1exp2^𝑇₀𝐻(𝑠,𝜔)𝑑𝐵_𝑠⩽𝐸^1/(1+2𝜖0)_𝑃𝜈1exp2+𝜖₀^𝑇₀𝐻(𝑠,𝜔)𝑑𝐵_𝑠⩽𝔼^1/(1+2𝜖0)1exp2+𝜖₀^𝑇₀𝐻(𝑠,𝜔)𝑑𝐵_𝑠<∞, (40) it follows from Kazamaki’s condition that ℰ(𝐵_𝑡) is a martingale under each 𝑃_𝜈, and 𝐸_𝑃𝜈[ℰ(𝐵_𝑡)]=1. Thus, 𝔼[ℰ(𝐵_𝑡)]=1, and ℰ(𝐵_𝑡) is symmetric.
We now claim that ℰ(𝐵_𝑡)∈𝕃¹_𝐺(ℱ_𝑡). By Lemma 5, it suffices to prove lim_𝑛→∞𝔼ℰ𝐵_𝑡𝐼_{{ℰ(𝐵𝑡)>𝑛}}=0. (41)
Now, 𝔼ℰ^1+𝑘𝐵_𝑡=𝔼exp(1+𝑘)^𝑡₀𝐻(𝑠,𝜔)𝑑𝐵_𝑠−1+𝑘2^𝑡₀𝐻²(𝑠,𝜔)𝑑⟨𝐵⟩_𝑠=𝔼√exp1+𝑘√1+𝑠^𝑡0𝐻(𝑠,𝜔)𝑑𝐵_𝑠−1+𝑘2^𝑡₀𝐻²(𝑠,𝜔)𝑑⟨𝐵⟩_𝑠)−√×exp(1+𝑘1+𝑘√1+𝑠^𝑡0𝐻(𝑠,𝜔)𝑑𝐵_𝑠⩽𝔼^{𝑠/(1+𝑠)}exp(1+𝑘)(1+𝑠)^𝑡₀𝐻(𝑠,𝜔)𝑑𝐵_𝑠−(1+𝑘)(1+𝑠)2^𝑡₀𝐻²(𝑠,𝜔)𝑑⟨𝐵⟩_𝑠×𝔼^{𝑠/(1+𝑠)}()−√exp1+𝑘1+𝑘√1+𝑠1+𝑠𝑠^𝑡₀𝐻(𝑠,𝜔)𝑑𝐵_𝑠=𝔼^{𝑠/(1+𝑠)}()−√exp1+𝑘1+𝑘√1+𝑠1+𝑠𝑠^𝑡₀𝐻(𝑠,𝜔)𝑑𝐵_𝑠, (42) where 𝑘 and 𝑠 are small positive numbers. Without loss of generality, let 𝑘=𝜃𝑠. Then we have lim_𝑠→0()−√1+𝑘1+𝑘√1+𝑠1+𝑠𝑠=lim_𝑠→0(1+𝜃𝑠)(1+𝑠)(1+𝜃𝑠)(1+𝑠)−1𝑠=12(1+𝜃). (43) So we can choose 𝜃 and 𝑠 small enough such that ()−√1+𝑘1+𝑘√1+𝑠1+𝑠𝑠<12+𝜀₀. (44) Then, 𝔼[ℰ^1+𝑘(𝐵_𝑡)]<∞ and 𝐶ℰ𝐵_𝑡⩽𝔼ℰ𝐵>𝑛^𝑡𝑛⟶0, (45) so that lim_𝑛→∞𝐸_𝐺ℰ𝐵_𝑡𝐼_{{ℰ(𝐵𝑡)>𝑛}}⩽𝔼^1/(1+𝑘)ℰ^1+𝑘𝐵_𝑡𝐶^{𝑘/(1+𝑘)}ℰ𝐵_𝑡>𝑛=0. (46) Since ℰ(𝐵_𝑡)∈𝐿¹_𝐺(ℱ_𝑡), and ℰ(𝐵_𝑡) is a martingale under each 𝑃_𝜈, we have 𝔼ℰ𝐵_𝑇∣ℱ_𝑡=𝐸_𝑃𝜈ℰ𝐵_𝑇∣ℱ_𝑡=ℰ𝐵_𝑡,𝑃_𝜈.a.s. (47) Then, 𝔼𝔼ℰ𝐵_𝑇∣ℱ_𝑡−ℰ𝐵_𝑡=sup_{𝑃𝜈∈𝒫}𝐸_𝑃𝜈𝔼ℰ𝐵_𝑇∣ℱ_𝑡−ℰ𝐵_𝑡=sup_{𝑃𝜈∈𝒫}𝐸_𝑃𝜈ℰ𝐵_𝑡−ℰ𝐵_𝑡=0, (48) which means that ℰ(𝐵_𝑡) is a symmetric martingale.

As a corollary, we can obtain the following criterion, because 𝔼1exp2+𝜀₀𝑀_∞⩽𝔼1exp2+𝜀₀⟨𝑀⟩_∞^1/2. (49)

Corollary 13. If there exists an 𝜀>0 such that 𝔼1exp2+𝜀₀^𝑇₀𝐻²(𝑠,𝜔)𝑑⟨𝐵⟩_𝑠<∞, (50) then ℰ(𝐵_𝑡) is a symmetric martingale under 𝐸_𝐺, and, for all 𝑡∈[0,𝑇],ℰ(𝐵_𝑡)∈𝐿¹_𝐺(ℱ_𝑡).

We will close this section with an example which tells a distinct difference between the above two criteria. Let 𝑀_𝑡=∫^𝑡₀𝐻(𝑠,𝜔)𝑑𝐵_𝑠.

Theorem 14. If ‖𝑀‖_∞<𝜋/2, then 𝔼1exp2+𝜀₀^𝑇₀𝐻²(𝑠,𝜔)𝑑⟨𝐵⟩_𝑠<∞. (51) However, there exists a 𝐺-martingale 𝑀 such that ‖𝑀‖_∞=𝜋/2 and 𝔼1exp2+𝜀₀^𝑇₀𝐻²(𝑠,𝜔)𝑑⟨𝐵⟩_𝑠=∞. (52)

To show this, one needs the next lemma.

Lemma 15 (see [7]). Let (𝑀)_𝑡⩾0 is a mean-square integrable martingale, 𝑎,𝑏>0 and 𝜏=inf{𝑡∶𝑀_𝑡∉(−𝑎,𝑏)}. Then one has 𝐸1exp2𝜃²⟨𝑀⟩_𝜏=cos(((𝑎−𝑏)/2)𝜃)))𝜃)𝜋cos(((𝑎+𝑏/20⩽𝜃<.𝑎+𝑏 (53)

Proof of Theorem 14. Firstly, let us consider the stopping time 𝜏=inf{𝑡∶𝐵_𝑡∉(−𝑑,𝑑)}, where 𝑑⩽(𝜋/2), and consider the process 𝑀=𝐵^𝜏. From [8], we know that (𝑀)_𝑡⩾0 is a 𝐺-symmetric martingale, and, under each probability measurement 𝑃∈𝒫,(𝑀)_𝑡⩾0 is a mean-square integrable martingale. Then, it follows from the lemma that 𝐸_𝑃1exp2⟨𝑀⟩_∞=𝐸_𝑃1exp2⟨𝐵⟩_𝜏=1𝜋cos𝑑<∞,𝑑<2lim_𝜃→11)𝜃)𝜋cos((𝜋/2=∞,𝑑=2; (54) that is, we use the relationship 𝔼[⋅]=sup_𝑃∈𝒫𝐸_𝑃[⋅] and complete the proof.

Acknowledgments

The project is sponsored by NSFC (11171062) and the Innovation Program of Shanghai Municipal Education Commission (12ZZ063).