Bernoulli

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

Extended Itô calculus for symmetric Markov processes

Alexander Walsh

Full-text: Open access

Abstract

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).

Article information

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

Dates
First available in Project Euclid: 12 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1352727805

Digital Object Identifier
doi:10.3150/11-BEJ377

Mathematical Reviews number (MathSciNet)
MR2995790

Zentralblatt MATH identifier
1278.60114

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

Citation

Walsh, Alexander. Extended Itô calculus for symmetric Markov processes. Bernoulli 18 (2012), no. 4, 1150--1171. doi:10.3150/11-BEJ377. https://projecteuclid.org/euclid.bj/1352727805


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