Advances in Pure Mathematics
Vol.04 No.10(2014), Article ID:51071,4 pages
10.4236/apm.2014.410063
Necessary and Sufficient Conditions for a Class Positive Local Martingale
Chuanzhong Chen, Saisai Yang
Department of Mathematics, Hainan Normal University, Haikou, China
Email: czchen@hainnu.edu.cn, yangsaisai1989@hotmail.com
Academic Editor: Zechun Hu, Department of Mathematics, Nanjing University, China
Copyright © 2014 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
Received 3 September 2014; revised 2 October 2014; accepted 13 October 2014
ABSTRACT
Let be a Markov process, which is assumed to be associated with a (non-symmetric) Dirichlet form on. 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).
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