On Itô's formula for elliptic diffusion processes

Xavier Bardina and Carles Rovira

Departament de Matemàtiques, Edifici C, Universitat Autònoma de Barcelona 08193 Bellaterra, Barcelona, Spain
Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007-Barcelona, Spain.

Abstract

In Bardina and Jolis (1997), the authors prove an extension of Itô's formula for $F(X_t,t)$, where $F(x,t)$ has a locally square integrable 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 formula was expressed using the quadratic covariation.

In the present paper, following the ideas of Eisenbaum (2000) for the Brownian motion, we show that one can re-express this formula using integration over space and time with respect to local times in place of quadratic covariation. We show also that when the function $F$ has a locally integrable derivative in $t$ we can avoid the mild continuity condition in $t$ for the derivative of $F$ in $x$.

Finally, we show that, under certain regularity on the coefficients, this Itô's formula can be applied to strongly elliptic and elliptic diffusions.

Keywords: Diffusion processes; Integration wrt local time; Itô's formula; Local time;

Published in: Bernoulli 13(3) (2007) 820-830