An extension of Itô's formula for elliptic diffusion processes.

Xavier Bardina and Maria Jolis

Departament de Matemàtiques, Edifici C, Universitat Autònoma de Barcelona
08193 Bellaterra, Barcelona, Spain

Abstract

We prove an extension of Itô's formula for $F(X_{t},t)$, where $F(x,t)$ has a locally square integrate derivative in $x$ that satisfies a mild continuity condition in $t$, and $X$ is a one-dimensional diffusion process such that the law of $X_{t}$ has a density satisfying some properties. Following the ideas of Föllmer, et al. (1995), where they prove an analogous extension when $X$ is the Brownian motion, the proof is based on the existence of a backward integral with respect to $X$. For this, conditions to ensure the reversibility of the diffusions property are needed.
In a second part of this paper we show, using techniques of Malliavin calculus, that, under certain regularity on the coefficients, the extended Itô's formula can be applied to strongly elliptic and elliptic diffusions.

Keywords: Itô's formula; Diffusion processes; Forward and backward integrals; Time reversal; Malliavin calculus

Published in: Stochastic processes and their applications, 69 (1997) 83-109.