On the Ito formula of noncausal type
Let \(B_ t\) be a real-valued Brownian motion and \(\{\phi_ n\}\) an orthonormal basis in \(L^ 2(0,1)\). Suppose f is a random function satisfying P(\(\int^{1}_{0}f^ 2(t)dt<\infty)=1\) and \(A\subseteq [0,1]\) a measurable set. If \[ \int_{A}fd_{\phi}B:=\lim \sum_{n}\int_{A}f(t,\omega)\phi_ n(t)dt\int^{1}_{0}\phi_ n(t)dB(t) \] converges in probability this expression is called a stochastic integral of noncausal type. The first author [Sur la question d'existence de solutions d'une équation différentielle stochastique du type noncausal. J. Math. Kyoto Univ. 24, 699-704 (1984)] considered stochastic integral equations of noncausal type. In this paper it is shown that the solution satisfies a formula of Ito's type in the non-causal case.
scientific article; zbMATH DE number 3897947