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.