Stochastic calculus for symmetric Markov processes

Abstract

Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It\^{o} formula for Dirichlet processes is obtained.Comment: Published in at http://dx.doi.org/10.1214/07-AOP347 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org); Errata DOI: 10.1214/11-AOP68

Perturbation of symmetric Markov processes

“…In this section, we give a quick review of Nakao's [17] definition of stochastic integral with respect to a additive functional of zero energy and our time-reversal approach of stochastic integral for Dirichlet processes developed in [3]. Let (N (x, dy), H t ) be a Lévy system for X; that is, N (x, dy) is a kernel on (E ∂ , B(E ∂ )) and H t is a PCAF with bounded 1-potential such that for any nonnegative Borel function φ on E ∂ × E ∂ vanishing on the diagonal and any…”

“…Nakao [17] has defined such kind of stochastic integral for a class of integrand but it is too restrictive for our investigation. In our recent paper [3], we established the needed stochastic integration theory for zero-energy additive functionals of X as well as the corresponding Itô formula via time-reversal technique. The main result of the current paper extends not only the results in [14] and [9] but also Feynman-Kac transforms by continuous additive functionals of zero energy studied in Chen and Zhang [8] and the pure-jump Girsanov transforms and discontinuous Feynman-Kac transforms in Chen [1] and in Chen and Song [5]- [6].…”

“…) is a PCAF, and the associated Revuz measure (as in (1.4)) is denoted by µ M (see [3]). More generally, if M u is the martingale part in the Fukushima decomposition (1.5) of u ∈ F, then M u , M is a CAF locally of bounded variation, and we have the associated Revuz measure µ M u ,M , which is locally the difference of smooth (positive) measures.…”

“…It requires to develop certain aspects of the stochastic calculus of symmetric Markov processes (in particular a general enough version of Itô's formula) to deal with these complications, which have been addressed very recently in our separate paper [3]. See section 2 below for a quick review.…”

“…An illuminating concrete example for Crucial to the development is to use the extension of Nakao's stochastic integral for zero-energy additive functionals and the associated Itô formula, both of which were recently developed in [3]. …”