  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.
