Estimation of the density of hypoelliptic
diffusion processes with application to an extended Itô's formula
Xavier Bardina and Maria Jolis
Departament de Matemàtiques,
Edifici C, Universitat Auṭnoma de Barcelona,
08193 Bellaterra, Barcelona, Spain
Abstract
We prove a uniform bound for the density, $p_t(x)$, of the
solution at time $t\in(0,1]$ of a 1-dimensional stochastic differential equation, under
hypoellipticity conditions. The same kind of bounds are obtained for
an expression involving the distributional derivative (with respect
to $x$) of $p_t(x)$. These results are applied to extend the Itô
formula to the composition of a function (satisfying slight
regularity conditions) with an hypoelliptic diffusion process in the
spirit of the work of Follmer, Protter and Shiryayev.
Published in: Journal of Theoretical Probability, 15 (1), 223-247, 2002