The Annals of Probability

Stochastic calculus over symmetric Markov processes without time reversal

Kazuhiro Kuwae

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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.

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

First available in Project Euclid: 8 July 2010

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Zentralblatt MATH identifier

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

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


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

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