2019/03/07 by Nacira Agram, Agram, Nacira, Bernt Øksendal +1
Computer Science · Economics, Econometrics and Finance · Mathematics · #60H05 #60H20 #60J75 #91B70 #91G80 #93E20 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Optimization and Control (math.OC) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1903.02936
openalex publication_date 2019/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (S% )^*. We show that this Hida-Malliavin derivative defined on L2(FT,P) is a natural extension of the classical Malliavin derivative defined on the subspace \mathbbD1,2 of L2(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 ∈ L2(FT,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.