An anticipatory Itô formula (Q1924876)

The authors extend the classical ``integration by parts'' formula, due to Itô for stochastic integrals with nonanticipative integrands, to certain anticipative integrands. This main announced result is the following Theorem: Let \(f\in L^\infty ([0,1])\) and \(X\) be a process given by the Wiener integral \(X(t)= \int^t_0 f(s)dB(s)\), \(t\in[0,1]\), where \(B\) is the Brownian motion. Let \(\theta:(x,y) \mapsto \mathbb{R}^2\) be a twice continuously differentiable function, \((x,y) \in\mathbb{R}^2\), such that the following processes \(\{\theta (X(\cdot), B(1))\), \({\partial^2 \theta \over \partial x^2} (X(\cdot)\), \(B(1))\), \({\partial^2 \theta\over \partial x\partial y} (X(\cdot), B(1))\}\) are in \(L^2([0,1] \times {\mathcal S}^1 (\mathbb{R}))\) where \({\mathcal S}^1(\mathbb{R})\) is the dual space of the real Schwartz space \({\mathcal S} (\mathbb{R})\). Then the integral \(\int^t_0 \partial^*_s(f(s)) {\partial\theta \over \partial x} (X(s), B(1))ds\) is a Hitsuda-Skorokhod integral, defined independently by both these authors in 1972 and 1975, respectively (and Hitsuda's name is often omitted in most of the recent research and the authors are correcting this oversight). Here \(\partial^*_t\) is the adjoint of the white noise differential operator \(\partial_t\) acting on \({\mathcal S} (\mathbb{R})\). Moreover, one has for \(0\leq t\leq 1\), \[ \begin{multlined} \theta \bigl(X(t),B(1)\bigr) =\theta \bigl(X(0), B(1)\bigr) +\int^t_0 \partial^*_s (f(s)) {\partial\theta \over\partial x} \bigl(X(s), B(1)\bigr)ds+ \\ +\int^t_0 \left[{1 \over 2} f(s)^2{\partial^2\theta\over\partial x^2}\bigl(X(s),B(1)\bigr)+f(s){\partial^2\theta\over\partial x \partial y} \bigl(X(s), B(1)\bigr) \right] ds. \end{multlined} \] The details of proof are given in the first author's book, ``White noise distribution theory'' (1996; Zbl 0853.60001). Some applications to stochastic integral equations are also sketched.