Extended Itô calculus for symmetric Markov processes

Abstract

Chen, Fitzsimmons, Kuwae and Zhang (Ann. Probab. 36 (2008) 931-970) have established an Ito 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 Ito formulas for semi-martingales using the local time stochastic calculus. But here the part of the local time is played by a process $({\Gamma }_t^a,a\in \mathbb{R},t\ge 0)$ defined thanks to Nakao's operator (Z. Wahrsch. Verw. Gebiete 68 (1985) 557-578).

Published in at http://dx.doi.org/10.3150/11-BEJ377 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (1)

• Absolute continuity of symmetric Markov processes