Functionals of Itô Processes as Stochastic Integrals

Abstract

Conditions are given under which a functional L of an Itô process $z( \cdot )$, \[(1)\qquad z(t) = z_0 + \int_0^t {f(s,z)ds} + \int_0^t {\sigma (s,z)\, dw} ,\quad 0 \leqq t \leqq 1,\] can be represented as \[L(z( \cdot ,w)) = \int_0^1 {\chi (t,w)dw} (t,w)\quad {\text{w.p. }}1,\] and an explicit formula for $\chi $ is given in terms of the Fréchet derivative of L and the solution of the linearized version of the Itô equation (1). The method of proof consists of applying a theorem of J. M. C. Clark to the Cauchy–Maruyama approximation of (1).