Construction of strong solutions of SDE's via Malliavin calculus (Q971842)

The authors aim at constructing strong solutions to stochastic differential equations with additive Wiener noise given by \(dX_t=b(t,X_t)dt+dB_t,\) where \(b\) is a Borel measurable function with some integrability conditions. Their new method of construction is mainly based on Malliavin calculus. They use it to directly construct a strong solution which together with uniqueness in law gives a unique strong solution which is Malliavin differentiable. This hints at a link between Malliavin differentiability and strong solutions of SDE. To be more precise, they apply a compactness criterion based on Malliavin calculus combined with local time variational calculus to an approximating sequence \(X_t^{(n)}\). The limit is a well defined element in the Hida distribution space satisfying the SDE via a transformation. They extend their result to time homogeneous Itô-diffusions and suggest their method to be expended to a broader class of noises, e.g. Lévy noise, fractional Brownian motion or even to SPDE.