Introduction to White Noise, Hida-Malliavin Calculus and Applications

Abstract

The purpose of these lectures is threefold: We first give a short survey of the Hida white noise calculus, and in this context we introduce the Hida-Malliavin derivative as a stochastic gradient with values in the Hida stochastic distribution space \((\mathcal{S}% )^*\). We show that this Hida-Malliavin derivative defined on \(L^2(\mathcal{F}_T,P)\) is a natural extension of the classical Malliavin derivative defined on the subspace \(\mathbb{D}_{1,2}\) of \(L^2(P)\). The Hida-Malliavin calculus allows us to prove new results under weaker assumptions than could be obtained by the classical theory. In particular, we prove the following: (i) A general integration by parts formula and duality theorem for Skorohod integrals, (ii) a generalised fundamental theorem of stochastic calculus, and (iii) a general Clark-Ocone theorem, valid for all \(F \in L^2(\mathcal{F}_T,P)\). As applications of the above theory we prove the following: A general representation theorem for backward stochastic differential equations with jumps, in terms of Hida-Malliavin derivatives; a general stochastic maximum principle for optimal control; backward stochastic Volterra integral equations; optimal control of stochastic Volterra integral equations and other stochastic systems.