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Introduction to White Noise, Hida-Malliavin Calculus and Applications
Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för matematik (MA). (Stochastic analysis and stochastic processes)
University of Oslo, Norway.
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
Abstract [en]

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 D1,2 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.

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Forskningsprogram
Matematik, Matematik
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URN: urn:nbn:se:lnu:diva-82305OAI: oai:DiVA.org:lnu-82305DiVA, id: diva2:1307466
Tilgjengelig fra: 2019-04-26 Laget: 2019-04-26 Sist oppdatert: 2019-12-20bibliografisk kontrollert

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