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November 2012 Extended Itô calculus for symmetric Markov processes
Alexander Walsh
Bernoulli 18(4): 1150-1171 (November 2012). DOI: 10.3150/11-BEJ377


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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Alexander Walsh. "Extended Itô calculus for symmetric Markov processes." Bernoulli 18 (4) 1150 - 1171, November 2012.


Published: November 2012
First available in Project Euclid: 12 November 2012

zbMATH: 1278.60114
MathSciNet: MR2995790
Digital Object Identifier: 10.3150/11-BEJ377

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

Rights: Copyright © 2012 Bernoulli Society for Mathematical Statistics and Probability


Vol.18 • No. 4 • November 2012
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