Malliavin Calculus for the one-dimensional Stochastic Stefan Problem

Dimitra C. Antonopoulou, Dimitrios Dimitriou, Georgia Karali, Konstantinos Tzirakis

Published 2024-07-29Version 1

We consider the one-dimensional outer stochastic Stefan problem with reflection which models the short-time prediction of the price or spread of one volatile asset traded in a financial market. The problem admits maximal solutions as long as the velocity of the moving boundary stays bounded, [3,6,7]. We apply Malliavin calculus on the transformed equation and prove first that its maximal solution u has continuous paths a.s. In the case of the unreflected problem, the previous enables localization of a proper approximating sequence of the maximal solution. Then, we derive there locally the differentiability of maximal $u$ in the Malliavin sense. The novelty of this work, apart from the derivation of continuity of the paths for the maximal solution with reflection, is that for the unreflected case we introduce a localization argument on maximal solutions and define efficiently the relevant sample space. More precisely, we present a methodology suitable for the proof of the expected local only (in the sample space) existence of the Malliavin derivative.