Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itos Formula

We prove a uniform bound for the density, p t (x), of the solution at time t(element of)(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p t (x). These results are applied to extend the Ito formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Follmer et al. (5)