An extension of Itoʹs formula for elliptic diffusion processes

We prove an extension of Itôʹs formula for F(Xt, t), where F(x, t) has a locally square integrable derivative in x that satisfies a mild continuity condition in t, and X is a one-dimensional diffusion process such that the law of Xt has a density satisfying some properties. Following the ideas of Fِllmer, 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 diffusion 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 Itôʹs formula can be applied to strongly elliptic and elliptic diffusions.