Estimation of the density of hypoelliptic diffusion processes with application to an extended Itô's formula

We prove a uniform bound for the density, pt(x), of the solution at time t ε (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 Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al.