Itô formula for an extended stochastic integral with nonanticipating kernel

Let $U_t =\int _0^1 u_s \mu (t,s)\delta W_s $ be an extended stochastic integral with a nonrandom anticipating kernel $\mu ( \,\cdot\, {,}\, \cdot\, )$. This paper gives the conditions of continuity for the process $U_t $ (§ 3), computes the quadratic variation (§ 4), and proves the Itô formula (§ 5) from which the formula for Brownian partial derivatives is deduced. With the help of the established Ito formula the probabilistic solution of some integro-differential equation is obtained (Example 3).