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