On the Ito formula for the Wiener process

The talk will describe how in the classical Ito formula to replace the second derivative, understood in the sense of ordinary differentiation, with the second derivative in the sense of differentiation of generalized functions. It will be shown that this can be done in the case when the first derivative belongs to the class $L_{2,loc}(R). Earlier, in the work of Felmer, Protter and Shiryaev, another form of the last term in the Ito formula was obtained under the same conditions.