• Bernoulli
  • Volume 18, Number 4 (2012), 1150-1171.

Extended Itô calculus for symmetric Markov processes

Alexander Walsh

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Chen, Fitzsimmons, Kuwae and Zhang (Ann. Probab. 36 (2008) 931–970) have established an Itô formula consisting in the development of $F(u(X))$ for a symmetric Markov process $X$, a function $u$ in the Dirichlet space of $X$ and any $\mathcal{C} ^{2}$-function $F$. We give here an extension of this formula for $u$ locally in the Dirichlet space of $X$ and $F$ admitting a locally bounded Radon–Nikodym derivative. This formula has some analogies with various extended Itô formulas for semi-martingales using the local time stochastic calculus. But here the part of the local time is played by a process $(\Gamma^{a}_{t},a\in \mathbb{R} ,t\geq0)$ defined thanks to Nakao’s operator (Z. Wahrsch. Verw. Gebiete 68 (1985) 557–578).

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Bernoulli, Volume 18, Number 4 (2012), 1150-1171.

First available in Project Euclid: 12 November 2012

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additive functional Fukushima decomposition Itô formula stochastic calculus symmetric Markov process zero energy process


Walsh, Alexander. Extended Itô calculus for symmetric Markov processes. Bernoulli 18 (2012), no. 4, 1150--1171. doi:10.3150/11-BEJ377.

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