A constructive approach to stochastic integral representations of non-smooth functionals of Brownian motion

O. Purtukhia

We study stochastic integral representations for a broad class of Brownian functionals that are not Malliavin-smooth. Classical Clark–Ocone formulas require the functional itself to be Malliavin differentiable and thus do not cover many objects of practical interest, particularly in mathematical finance. We first consider functionals that, while not Malliavin differentiable, have conditional expectations with respect to the Brownian filtration that do admit a Malliavin derivative. For this class, we derive a generalized Clark–Ocone representation with an explicit, constructive integrand. We then address genuinely irregular path-dependent functionals whose conditional expectations are also non-Malliavin differentiable; under suitable structural conditions, stochastic integral representations with explicit integrands are still obtained. These results significantly expand the class of Brownian functionals admitting constructive stochastic integral representations. This work extends our earlier studies (including joint work with Glonti, Jaoshvili, and others); to ensure self-containment, we provide a brief review. The first section introduces stochastic integral representations and stochastic derivatives, reviews results from Itô’s representation onward, and presents the Kallianpur counterexample. The second section develops the Glonti–Purtukhia generalization of the Clark–Ocone formula and its applications. The third section focuses on integral representations of path-dependent Brownian functionals. 

Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 19(2) (2026), pp. 53-70