An extension of Itô's formula for elliptic diffusion processes
Recently, \textit{H. Föllmer, P. Protter} and \textit{A. N. Shiryayev} extended Itô's formula to processes of the form \(F(X_{t})\), where \(F\) is an absolutely continuous function on \(\mathbb R\) with locally square integrable derivative and \(X\) is a standard one-dimensional Wiener process [see Bernoulli 1, No. 1/2, 149-169 (1995; Zbl 0851.60048)]. In the present paper, their result is further generalized, \(X_{t}\) being assumed now to be a one-dimensional diffusion process solving a stochastic differential equation \[ dX_{t} = b(t,X_{t}) dt + \sigma (t,X_{t}) dW_{t} \] with Lipschitz continuous coefficients of a linear growth and such that \(X_{t}\) has a density \(p_{t}(x)\) satisfying some integrability conditions. Functions \(F\) depending on \(t\) are allowed as well. In the second part of the paper the authors find, using the Malliavin calculus, sufficient conditions (in terms of the non-degeneracy of \(\sigma \) and regularity of \(b\) and \(\sigma \)) for the assumptions on \(p_{t}\) to hold.
scientific article; zbMATH DE number 1240028
scientific article