DIFFERENT APPROACHES IN THE CONSTRUCTIVE MARTINGALE REPRESENTATION OF BROWNIAN FUNCTIONALS

E. B. Namgalauri; O. G. Purtukhia

doi:10.17721/2706-9699.2022.1.03

E. B. Namgalauri Faculty of Exact and Natural Sciences, Department of Mathematics, Ivane Javakhishvili Tbilisi State University, Tbilisi, Georgia
O. G. Purtukhia Faculty of Exact and Natural Sciences, Department of Mathematics and A. Razmadze Mathematical Institute, Ivane Javakhishvili Tbilisi State University, Tbilisi, Georgia

DOI: https://doi.org/10.17721/2706-9699.2022.1.03

Keywords: martingale representation, Malliavin derivative, Clark– Ocone formula, Glonti–Purtukhia representation

Abstract

In this work, we study the issues of a constructive stochastic integral representation of Brownian functionals, which are interesting from the point of view of their practical application in the problem of hedging a European option. In addition to briefly discussing known results in this direction, in the case of stochastically smooth (in Malliavin sense) functionals, we also illustrate the usefulness of the Glonti–Purtukhia representation for non-smooth functionals. In particular, we generalize the Clarke–Ocone formula to the case when the functional is not stochastically smooth, but its conditional mathematical expectation is stochastically differentiable, and find an explicit expression for the integrand. Moreover, we consider such functionals that do not satisfy even weakened conditions, that is, non-smooth, past-dependent Brownian functionals, the conditional mathematical expectations of which are also not stochastically differentiable, and again we give a constructive martingale representation.

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