The Annals of Probability

Stochastic calculus over symmetric Markov processes without time reversal

Kazuhiro Kuwae

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Abstract

We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakao’s divergence-like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi-everywhere starting points, which are refinements of the previous results under the law for almost everywhere starting points. This refinement of stochastic calculus enables us to establish a generalized Fukushima decomposition for a certain class of functions locally in the domain of Dirichlet form and a generalized Itô formula.

Article information

Source
Ann. Probab., Volume 38, Number 4 (2010), 1532-1569.

Dates
First available in Project Euclid: 8 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1278593959

Digital Object Identifier
doi:10.1214/09-AOP516

Mathematical Reviews number (MathSciNet)
MR2663636

Zentralblatt MATH identifier
1206.31009

Subjects
Primary: 31C25: Dirichlet spaces
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J75: Jump processes

Keywords
Symmetric Markov process Dirichlet form Revuz measure martingale additive functionals of finite energy continuous additive functional of zero energy Nakao’s CAF of zero energy Fukushima decomposition semi-martingale Dirichlet processes stochastic integral Itô integral Fisk–Stratonovich integral time reversal operator dual predictable projection

Citation

Kuwae, Kazuhiro. Stochastic calculus over symmetric Markov processes without time reversal. Ann. Probab. 38 (2010), no. 4, 1532--1569. doi:10.1214/09-AOP516. https://projecteuclid.org/euclid.aop/1278593959


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