One remark to the Itô formula

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.