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:
(1)∂∂tu(t,x)=G(Δu),(t,x)∈[0,+∞)×ℝ,u(0,x)=φ(x),
where Δ is Laplacian and the sublinear function G is defined by
(2)G(α)=12(σ¯2α+-σ_2α-),α∈ℝ,
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 Bt+s-Bt being G-normally distributed under a given sublinear expectation𝔼^. 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
(3)𝔼^[exp{(12+ε0)∫0TH(s,ω)dBs}]<∞,
then
(4)ℰ(Bt):=exp{∫0tH(s,ω)dBs-12∫0tH2(s,ω)d〈B〉s}
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
(5)Eexp(12M∞)<∞,
then the martingale
(6)exp{Mt-12〈M〉t},0≤t≤∞,
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

monotonicity: if X≥Y, then 𝔼^[X]≥𝔼^[Y];

constant preserving: 𝔼^[c]=c, for all c∈ℝ;

subadditivity:𝔼^[X]-𝔼^[Y]≤𝔼^[X-Y];

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
(7)φ(X1,…,Xd)∈ℋ
if Xi∈ℋ,i=1,…,d, for all φ∈Cb,Lip(ℝd), where Cb,Lip(ℝd) denotes the space of all bounded and Lipschitz functions on ℝd. In a sublinear expectation space (Ω,ℋ,𝔼^), a random vector Y=(Y1,…,Yn),Yi∈ℋ, is said to be independent under 𝔼^ from another random vector X=(X1,…,Xm),Xi∈ℋ, if for each test function φ∈Cb,Lip(ℝm+n), one has
(8)𝔼^[φ(X,Y)]=𝔼^[𝔼^[φ(x,Y)]x=X].
Two n-dimensional random vectors X and Y defined, respectively, in the sublinear expectation spaces (Ω1,ℋ1,𝔼^1) and (Ω2,ℋ2,𝔼^2) are called identically distributed, denoted by X~Y, if
(9)𝔼^1[φ(X)]=𝔼^2[φ(Y)],
for all φ∈Cb,Lip(ℝn).

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,[σ_2,σ¯2]), if for each φ∈Cb,Lip(ℝ), the function defined by
(10)u(t,x):=𝔼^[φ(x+tξ)],(t,x)∈[0,∞)×ℝ,
is the unique viscosity solution of the following nonlinear heat equation:
(11)∂∂tu(t,x)=G(Δu),(t,x)∈[0,+∞)×ℝ,u(0,x)=φ(x),
where Δ is Laplacian and the sublinear function G is defined by
(12)G(α)=12(σ¯2α+-σ_2α-),α∈ℝ.

Example 3 (Peng [<xref ref-type="bibr" rid="B5">2</xref>]).

Let ξ~N(0,[σ_2,σ¯2]). We then have
(13)𝔼^[φ(ξ)]=12πσ¯∫ℝφ(x)e(-1/2σ¯2)x2dx
for all convex functions φ and
(14)𝔼^[ψ(ξ)]=12πσ_∫ℝφ(x)e(-1/2σ_2)x2dx
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
(15)ρ(ω1,ω2)=∑i=1∞2-i[(maxt∈[0,i]|ωt1-ωt2|)∧1],ω1,ω2∈Ω.
We denote by ℬ(Ω) the Borel-algebra on Ω. We also denote, for each t∈[0,∞),
(16)Ωt={ω·∧t,ω∈Ω}
and ℱt=ℬ(Ωt), where x∧y=min{x,y}. We also denote the following:

L0(Ω): the space of all ℬ(Ω)-measurable real valued functions on Ω;

L0(Ωt): the space of all ℬ(Ωt)-measurable real valued functions on Ωt;

Lb(Ω): the space of all bounded elements in L0(Ω);

Lb(Ωt): the space of all bounded elements in L0(Ωt).

Let 𝕃Gp(Ω) be the closure of ℋ with respect to the norm
(17)∥X∥p=𝔼^[|X|p]1/p
with p∈[1,∞). Clearly, the space 𝕃Gp(Ω) is a Banach space and the space Cb(Ω) of bounded continuous functions on Ω is a subset of 𝕃G1(Ω), and, moreover, for the sublinear expectation space (Ω,𝕃Gp(Ω),𝔼^), there exists a weakly compact family 𝒫 of probability measures on (Ω,ℬ(Ω)) such that
(18)𝔼^=supP∈𝒫EP.
So we can introduce the Choquet capacity C^ by taking
(19)C^(A)=supP∈𝒫P(A),A∈ℬ(Ω).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 𝕃Gp(Ω) as
(20)𝕃Gp(Ω)={L0(Ω)∋Xiscontinuous,q.s.,supP∈𝒫EP[|X|p]hjhsupP∈𝒫EP[|X|p]<∞}≡{L0(Ω)∋Xiscontinuous,q.s.,limn→∞supP∈𝒫EP[|X|p1{|X|>n}]fflimn→∞supP∈𝒫EP[|X|p1{|X|>n}]=0}.
The following three results can be consulted in Denis et al. [4] and Hu and Peng [5].

Lemma 5 (Denis et al. [<xref ref-type="bibr" rid="B1">4</xref>] and Hu and Peng [<xref ref-type="bibr" rid="B2">5</xref>]).

Let {Xn,n=1,2,…} be a monotonically decreasing sequence of nonnegative random variances in Cb(Ω). If Xn converges to zero q.s. on Ω, then one has
(21)limn→0𝔼^[Xn]=0.
Moreover, if Xn↑X and 𝔼^[X] and 𝔼^[Xn] are finite for all n=1,2,…, one then has
(22)limn→0𝔼^[Xn]=𝔼^[X].

Lemma 6 (Denis et al. [<xref ref-type="bibr" rid="B1">4</xref>] and Hu and Peng [<xref ref-type="bibr" rid="B2">5</xref>]).

Let 1≤p<∞. Consider the sets 𝕃Gp(Ω) and 𝕃p=ℒp/𝒩, where
(23)ℒp={X∈L0(Ω):𝔼^(|X|p)=supP∈𝒫EP[|X|p]<∞},𝒩={X∈L0(Ω):X=0q.s.}.
Then,

𝕃p is a Banach space with respect to the norm ∥·∥p;

𝕃Gp is the completion of Cb(Ω) with respect to the norm ∥·∥p.

Lemma 7 (Denis et al. [<xref ref-type="bibr" rid="B1">4</xref>] and Hu and Peng [<xref ref-type="bibr" rid="B2">5</xref>]).

For a given p∈(0,+∞], if the sequence 𝕃p⊃{Xn} converges to X in 𝕃p, then there exists a subsequence {Xnk} such that Xnk converges to X quasi-surely.

We denote by 𝕃*p(Ω) the completion of Lb(Ω) with respect to the norm ∥·∥p.

A process B={Bt,t≥0}⊂ℋ in a sublinear expectation space (Ω,ℋ,𝔼^) is called a G-Brownian motion if the following properties are satisfied:

B0=0;

for each t,s≥0, the increment Bt+s-Bt is N(0,[σ_2s,σ¯2s])-distributed and is independent from (Bt1,…,Btn), for all n=0,1,2,… and 0≤t1≤t2≤⋯≤tn≤t.

The G-Brownian motion B has the following properties:

for all ξ∈𝕃2(Ωt), one has 𝔼^[ξ(BT-Bt)]=0 with 0≤t≤T;

for all ℬ(Ωt)-measurable real valued, bounded functions ξ, one has
(24)𝔼^[ξ2(BT-Bt)2]≤σ¯2(T-t)𝔼^[ξ2],0≤t≤T;

for all t≥0, one has 𝔼^[Bt]=𝔼^[-Bt]=0;

t↦Bt 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 Mbp,0([0,T]) the set of step processes as follows:
(25)ηt(ω)=∑j=1Nξj(ω)1[tj-1,tj)(t),ξj∈Lb(Ωtj-1)
with 0=t0<⋯<tN=T. For the process of the form (25) one defines the related Bochner integral as follows:
(26)∫0Tηtdt=∑j=1Nξj(tj-tj-1).
For every η∈Mbp,0([0,T]), one sets
(27)𝔼^T(η):=1T𝔼^∫0Tηtdt.
Then 𝔼^T forms a sublinear expectation. Moreover, one denotes by M*p([0,T]) the completion of Mbp,0([0,T]) under the norm
(28)∥η∥M*p([0,T])=(𝔼^[∫0T|ηs|pds])1/p.

Definition 9.

For every η∈Mbp,0([0,T]) of the form (25), one defines the Itô integral of η with respect to G-Brownian motion B by
(29)I(η):=∫0TηsdBs=∑j=1Nξj(Btj-Btj-1).

The mapping I:Mbp,0([0,T])→𝕃*2(ΩT) is a linear continuous mapping and thus can be continuously extended to I:M*2([0,T])→𝕃*2(ΩT), which is called the Itô integral of η∈M*2([0,T]) with respect to G-Brownian motion B, and define
(30)∫0tηsdBs=∫0T1{0≤s≤t}ηsdBs
for all η∈M*2([0,T]) and t∈[0,T]. One has
(31)𝔼^(∫0TηsdBs)=0,𝔼^[(∫0TηsdBs)2]≤σ¯2𝔼^[∫0Tηs2ds]
for all η∈M*2([0,T]). Moreover, the process {∫0tηsdBs,t∈[0,T]} is continuous in t quasi-surely and
(32)∫0·ηsdBs∈M*2([0,T])
for all η∈M*2([0,T]).

Definition 10 (quadratic variation).

Let πtN={0=t0N<t1N<⋯<tN-1N=t} be a partition of [0,t] for t>0, such that μ(πtN):=maxj{tj-tj-1}→0 as N→∞. The quadratic variation of G-Brownian motion B is defined by
(33)〈B〉t=limμ(πtN)→0∑k=0N-1(Btk+1N-BtkN)2=Bt2-2∫0tBsdBs
in 𝕃G2(Ω).

The function t↦〈B〉t is continuous and increasing outside a polar set. One can define the integral
(34)∫0Tηtd〈B〉t:=∑j=1Nξj(〈B〉tj-〈B〉tj-1)
as a map from Mb1,0([0,T]) into 𝕃*1(ΩT), and the map is linear and continuous, and it can be extended continuously to M*1([0,T]).

Definition 11.

A process (Mt)t⩾0 is called a G-martingale if, for each t∈[0,∞),Mt∈𝕃G1(ℱt) and, for each s∈[0,t], one has
(35)𝔼^[Mt∣ℱs]=Ms.
If both (Mt)t⩾0 and (-Mt)t⩾0 are G-martingales, (Mt)t⩾0 is called a G-symmetric martingale.

One can easily prove that the process
(36)Mt:=𝔼^[X∣ℱt],t≥0,
is a G-martingale for each X∈𝕃G1(ℱ), and, moreover, the process
(37)Mt≔M0+∫0tφudBu+∫0tηd〈B〉u-∫0t2G(ηu)du,∀t∈[0,T]
also is a G-martingale for all M0∈ℝ,φ∈Mb2[0,T],η∈Mb1[0,T].

3. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>Theorem 12.

If there exists an ϵ>0 such that
(38)𝔼^[exp{(12+ϵ0)∫0TH(s,ω)dBs}]<∞,
then
(39)ℰ(Bt)≔exp{∫0tH(s,ω)dBs-12∫0tH2(s,ω)d〈B〉s}
is a symmetric martingale under 𝔼^, and, for all t∈[0,T],ℰ(Bt)∈𝕃G1(ℱt).

Proof.

Since Bt is a mean-square integrable martingale under each Pν and
(40)EPν[exp{12∫0TH(s,ω)dBs}]⩽EPν1/(1+2ϵ0)[exp{(12+ϵ0)∫0TH(s,ω)dBs}]⩽𝔼^1/(1+2ϵ0)[exp{(12+ϵ0)∫0TH(s,ω)dBs}]<∞,
it follows from Kazamaki’s condition that ℰ(Bt) is a martingale under each Pν, and EPν[ℰ(Bt)]=1. Thus, 𝔼^[ℰ(Bt)]=1, and ℰ(Bt) is symmetric.

We now claim that ℰ(Bt)∈𝕃G1(ℱt). By Lemma 5, it suffices to prove
(41)limn→∞𝔼^[ℰ(Bt)I{ℰ(Bt)>n}]=0.

Now,
(42)𝔼^[ℰ1+k(Bt)]=𝔼^[exp{(1+k)∫0tH(s,ω)dBs-1+k2∫0tH2(s,ω)d〈B〉s}]=𝔼^[exp{1+k1+s∫0tH(s,ω)dBs-1+k2∫0tH2(s,ω)d〈B〉s}jjjjjj×exp{((1+k)-1+k1+s)∫0tH(s,ω)dBs}]⩽𝔼^s/(1+s)[exp{(1+k)(1+s)∫0tH(s,ω)dBs⩽1𝔼^11+s-(1+k)(1+s)2∫0tH2(s,ω)d〈B〉s}]×𝔼^s/(1+s)[exp{((1+k)-1+k1+s)1+ss∫0tH(s,ω)dBs}]=𝔼^s/(1+s)[exp{((1+k)-1+k1+s)1+ss∫0tH(s,ω)dBs}],
where k and s are small positive numbers. Without loss of generality, let k=θs. Then we have
(43)lims→0((1+k)-1+k1+s)1+ss=lims→0(1+θs)(1+s)((1+θs)(1+s)-1)s=12(1+θ).
So we can choose θ and s small enough such that
(44)((1+k)-1+k1+s)1+ss<12+ε0.
Then, 𝔼^[ℰ1+k(Bt)]<∞ and
(45)C^[ℰ(Bt)>n]⩽𝔼^[ℰ(Bt)]n⟶0,
so that
(46)limn→∞EG[ℰ(Bt)I{ℰ(Bt)>n}]⩽𝔼^1/(1+k)[ℰ1+k(Bt)]C^k/(1+k)[ℰ(Bt)>n]=0.
Since ℰ(Bt)∈LG1(ℱt), and ℰ(Bt) is a martingale under each Pν, we have
(47)𝔼^[ℰ(BT)∣ℱt]=EPν[ℰ(BT)∣ℱt]=ℰ(Bt),Pν.a.s.
Then,
(48)𝔼^[|𝔼^[ℰ(BT)∣ℱt]-ℰ(Bt)|]=supPν∈𝒫EPν[|𝔼^[ℰ(BT)∣ℱt]-ℰ(Bt)|]=supPν∈𝒫EPν[|ℰ(Bt)-ℰ(Bt)|]=0,
which means that ℰ(Bt) is a symmetric martingale.

As a corollary, we can obtain the following criterion, because
(49)𝔼^{exp[(12+ε0)M∞]}⩽𝔼^{exp[(12+ε0)〈M〉∞]}1/2.

Corollary 13.

If there exists an ε>0 such that
(50)𝔼^[exp{(12+ε0)∫0TH2(s,ω)d〈B〉s}]<∞,
then ℰ(Bt) is a symmetric martingale under EG, and, for all t∈[0,T],ℰ(Bt)∈LG1(ℱt).

We will close this section with an example which tells a distinct difference between the above two criteria. Let Mt=∫0tH(s,ω)dBs.

Theorem 14.

If ∥M∥∞<π/2, then
(51)𝔼^[exp{(12+ε0)∫0TH2(s,ω)d〈B〉s}]<∞.
However, there exists a G-martingale M such that ∥M∥∞=π/2 and
(52)𝔼^[exp{(12+ε0)∫0TH2(s,ω)d〈B〉s}]=∞.

To show this, one needs the next lemma.

Lemma 15 (see [<xref ref-type="bibr" rid="B3">7</xref>]).

Let (M)t⩾0 is a mean-square integrable martingale, a,b>0 and τ=inf{t:Mt∉(-a,b)}. Then one has
(53)E[exp(12θ2〈M〉τ)]=cos(((a-b)/2)θ)cos(((a+b)/2)θ)(0⩽θ<πa+b).

Proof of Theorem <xref ref-type="statement" rid="thm3.2">14</xref>.

Firstly, let us consider the stopping time τ=inf{t:Bt∉(-d,d)}, where d⩽(π/2), and consider the process M=Bτ. 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
(54)EP[exp(12〈M〉∞)]=EP[exp(12〈B〉τ)]={1cosd<∞,d<π2limθ→11cos((π/2)θ)=∞,d=π2;
that is, we use the relationship 𝔼^[·]=supP∈𝒫EP[·] and complete the proof.

Acknowledgments

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

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