Abstract and Applied Analysis

Volume 2013 (2013), Article ID 247307, 6 pages

http://dx.doi.org/10.1155/2013/247307

Research Article

A Note on the Exponential G-Martingale

Yunsheng Lu1,2 and Yingying Liu²

¹Glorious Sun School of Business and Management, Donghua University, 1882 West Yanan Road, Shanghai 200051, China

²Department of Mathematics, Donghua University, 2999 North Renmin Road, Songjiang, Shanghai 201620, China

Received 14 November 2013; Accepted 26 November 2013

Academic Editor: Litan Yan

Copyright © 2013 Yunsheng Lu and Yingying Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 -expectation (see also Peng [2]), which is associated with the following nonlinear heat equation:

where is Laplacian and the sublinear function is defined by

with two given constants . Together with the notion of -expectations, Peng also introduced the related -normal distribution, the -Brownian motion, and related stochastic calculus under -expectation, and moreover an Itô's formula for the -Brownian motion was established. -Brownian motion has a very rich and interesting new structure which nontrivially generalizes the classical one. Briefly speaking, a -Brownian motion is a continuous process with independent stationary increments being -normally distributed under a given sublinear expectation  . A very interesting new phenomenon of -Brownian motion is that its quadratic process 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 -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 such that

then

is a symmetric martingale under .

Under the classical case, the result is called Kazamaki's condition, and it can be recalled as follows, for any classical continuous martingale , if

then the martingale

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 -Brownian motion and -Expectation. In Section 3 we prove the above theorem and discuss some examples.

2. Preliminaries

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

Let be a given set and let be a linear space of real valued functions defined on such that and for all .

Definition 2. A sublinear expectation on is a functional with the following properties, for all , one has

(i)monotonicity: if , then ;(ii)constant preserving:  , for all ;(iii)subadditivity:;(iv)positive homogeneity: , for all .

The triple is called a sublinear expectation space, and is considered as the space of random variables on .

It is important to note that one can suppose that

if , for all , where denotes the space of all bounded and Lipschitz functions on . In a sublinear expectation space , a random vector , is said to be independent under from another random vector , if for each test function , one has

Two -dimensional random vectors and defined, respectively, in the sublinear expectation spaces and are called identically distributed, denoted by , if

for all .

Let and be two real numbers with . A random variable in a sublinear expectation space is called -normal distributed, denoted by , if for each , the function defined by

is the unique viscosity solution of the following nonlinear heat equation:

where is Laplacian and the sublinear function is defined by

Example 3 (Peng [2]). Let . We then have

for all convex functions and

for all concave functions .

Let now be the space of all real valued continuous functions on with initial value , equipped with the distance

We denote by the Borel-algebra on . We also denote, for each ,

and , where . We also denote the following:

(i): the space of all -measurable real valued functions on ;(ii): the space of all -measurable real valued functions on ;(iii): the space of all bounded elements in ;(iv): the space of all bounded elements in .

Let be the closure of with respect to the norm

with . Clearly, the space is a Banach space and the space of bounded continuous functions on is a subset of , and, moreover, for the sublinear expectation space , there exists a weakly compact family of probability measures on such that

So we can introduce the Choquet capacity by taking

Definition 4. A set is called polar if . 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 , one can characterize the space as

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 be a monotonically decreasing sequence of nonnegative random variances in . If converges to zero q.s. on , then one has

Moreover, if and and are finite for all , one then has

Lemma 6 (Denis et al. [4] and Hu and Peng [5]). Let . Consider the sets and , where

Then,

(i) is a Banach space with respect to the norm ;(ii) is the completion of with respect to the norm .

Lemma 7 (Denis et al. [4] and Hu and Peng [5]). For a given , if the sequence converges to in , then there exists a subsequence such that converges to quasi-surely.

We denote by the completion of with respect to the norm .

Definition 8 (-Brownian motion). A process in a sublinear expectation space is called a -Brownian motion if the following properties are satisfied:

(i);(ii)for each , the increment is -distributed and is independent from , for all and .

The -Brownian motion has the following properties:

(1)for all , one has with ;(2)for all -measurable real valued, bounded functions , one has

(3)for all , one has ;(4) is Hölder continuous of order , quasi-surely.

In Li and Peng [6], a generalized Itô integral and a generalized Itô formula with respect to the -Brownian motion are introduced. For arbitrarily fixed and , one denotes by the set of step processes as follows:

with . For the process of the form (25) one defines the related Bochner integral as follows:

For every , one sets

Then forms a sublinear expectation. Moreover, one denotes by the completion of under the norm

Definition 9. For every of the form (25), one defines the Itô integral of with respect to -Brownian motion by

The mapping is a linear continuous mapping and thus can be continuously extended to , which is called the Itô integral of with respect to -Brownian motion , and define

for all and . One has

for all . Moreover, the process is continuous in quasi-surely and

for all .

Definition 10 (quadratic variation). Let be a partition of for , such that as . The quadratic variation of -Brownian motion is defined by

in .

The function is continuous and increasing outside a polar set. One can define the integral

as a map from into , and the map is linear and continuous, and it can be extended continuously to .

Definition 11. A process is called a -martingale if, for each and, for each , one has

If both and are -martingales, is called a -symmetric martingale.

One can easily prove that the process

is a -martingale for each , and, moreover, the process

also is a -martingale for all .

3. Proof of Theorem 1

Theorem 12. If there exists an such that

then

is a symmetric martingale under , and, for all .

Proof. Since is a mean-square integrable martingale under each and

it follows from Kazamaki’s condition that is a martingale under each , and . Thus, , and is symmetric.

We now claim that . By Lemma 5, it suffices to prove

Now,

where and are small positive numbers. Without loss of generality, let . Then we have

So we can choose and small enough such that

Then, and

so that

Since , and is a martingale under each , we have

Then,

which means that is a symmetric martingale.

As a corollary, we can obtain the following criterion, because

Corollary 13. If there exists an such that

then is a symmetric martingale under , and, for all .

We will close this section with an example which tells a distinct difference between the above two criteria. Let .

Theorem 14. If , then

However, there exists a -martingale such that and

To show this, one needs the next lemma.

Lemma 15 (see [7]). Let is a mean-square integrable martingale, and . Then one has

Proof of Theorem 14. Firstly, let us consider the stopping time , where , and consider the process . From [8], we know that is a -symmetric martingale, and, under each probability measurement is a mean-square integrable martingale. Then, it follows from the lemma that

that is, we use the relationship and complete the proof.

Acknowledgments

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

References

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