A Remark on the Itô Formula

Abstract

Theory of Probability &Its Applications, Volume 69, Issue 2, Page 227-242, August 2024.
In the classical Itô formula, we propose replacing the second derivative (understood in the usual sense) by the second derivative in the sense of differentiation of distributions. In particular, we show that this can be done if the first derivative lies in the class $L_{2,\mathrm{loc}}(\mathbf{R})$. Earlier, Föllmer, Protter, and Shiryayev [Bernoulli, 1 (1995), pp. 149--169] obtained a different form of the last term in the Itô formula under the same conditions.