Necessary and Sufficient Conditions for a Class Positive Local Martingale

Chuanzhong Chen, Saisai Yang^*

Department of Mathematics, Hainan Normal University, Haikou, China.

DOI: 10.4236/apm.2014.410063                

Abstract

Let X be a Markov process, which is assumed to be associated with a (non-symmetric) Dirichlet form (E,D(E)) on L² (E;m). For , the extended Dirichlet space, we give necessary and sufficient conditions for a multiplicative functional to be a positive local martingale.

Keywords

Markov Process, Dirichlet Form, Multiplicative Functional, Positive Local Martingale

1. Introduction

Let be a (non-symmetric) Markov process on a metrizable Lusin space and be a -finite positive measure on its Borel -algebra. Suppose that is a quasi-regular Dirichlet form on associated with Markov process (we refer the reader to [1] [2] for notations and terminologies of this paper). To simplify notation, we will denote by its -quasi- continuous -version. If, then there exist unique martingale additive functional (MAF in short) of finite energy and continuous additive functional (CAF in short) of zero energy such that

Let be a Lévy system for and be the Revuz measure of the positive continuous additive functional (PCAF in short). For, we define the -valued functional

This paper is concerned with the following multiplicative functionals for:

(1)

where is the sharp bracket PCAF of the continuous part of.

In [3] under the assumption that is a diffusion process, then is a positive local martin-

gale and hence a positive supermartingale. In [4] , under the assumption that is bounded or, it is shown that is a positive local martingale and hence induces another Markov process, which is called the Girsanov transformed process of (see [5] ). Chen et al. in [5] give some necessary and sufficient conditions for to be a positive supermartingale when the Markov processes are symmetric. It is worthy to point out that the Beurling-Deny formula and Lyons-Zheng decomposition do not apply well to non- symmetric Dirichlet forms setting. For the non-symmetric situations, , an interesting and important question is that under what condition is a positive local martingale?

In this paper, we will try to give a complete answer to this question when the Dirichlet forms are non-sym- metric. We present necessary and sufficient conditions for to be a positive local martingale.

2. Main Result

Recall that a positive measure on is called smooth with respect to if whenever is -exceptional and there exists an -nest of compact subsets of E such that

Let, , We know from [6] that J, k are Randon measures.

Let, be defined as in (1). Denote

Now we can state the main result of this paper.

Theorem 1 The following are equivalent:

(i) is a positive -local martingale on for.

(ii) is locally -integrable on for.

(iii) is a smooth measure on.

Proof. (iii) (ii) Suppose that is a smooth measure on and is an -nest such

that and is of finite energy integral for. Similar to Lemma 2.4 of [4] ,

is quasi-continuous and hence finite. Denote. Then for

,

Hence by proposition IV 5.30 of [1] is locally -integrable on for.

(ii) (i) Assume that is locally -integrable on for. One can check that for

the dual predictable projection of on is. We set

Then is a local martingale on and the solution of the stochastic differential equation (SDE)

is a local martingale on. Moreover, by Doleans-Dade formula (cf. 9.39 of [7] ), Note that, we have that

So is a -local martingale.

Let. Note that is a càdlàg process, there are at most countably

many points at which. Since by Lemma 7.27 of [7] -, there are

only finitely many points at which, which give a finite non-zero contribution to the prod-

uct. Using the inequality when, we get

Therefore is a positive -local martingale on for.

(i) (iii) Assume that is a positive -local martingale on for, by Lemma 2.2 and Lemma 2.4 of [8] ,

is a local martingale on. We set

then is also a local martingale on. Denote is the

purely discontinuous part of, by Theorem 7.17 of [7] , there exist a locally bounded martingale and a local martingale of integrable variation such that. Since is -quasi-continuous, take

an -nest consisting of compact sets such that and is continuous hence

bounded for each. Denote

Take a,. Set, where is the family of resolvents associated with

. Since is dense in the -norm, by proposition III. 3.5 and 3.6 of

[1] , there exists an -nest consisting of compact sets and a sequence such

that, for some and converges to uniformly on as

for each. Set. So there exists an non-negative and constant

such that on. Suppose, then

where denotes the supremum norm. Recall that a locally bounded martingale is a locally square integrable martingales, is a locally square integrable martingales and is a local martingale of integrable variation. Therefore the quadratic variation is -locally integrable for, hence there exist a predictable dual projection which is a CAF of finite variation. Since

the Revuz measure of is

Let be a generalized -nest associated with such that for each. Denote, then and is an -nest. Hence for any, we have. On the other hand, as is bounded, there exists a positive constant such that and are not larger than. Because are Radon measure and are bounded,

As inequality on and on, we have for any non-negative,

For is an -nest consisting of compact sets, similar to, we can construct an -nest

consisting of compact sets such that for each. And there exists a sequence non-

negative such that on for each and some positive.

Since, is a smooth measure on.

Acknowledgments

We are grateful to the support of NSFC (Grant No. 10961012).

Conflicts of Interest

The authors declare no conflicts of interest.

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