Construction of strong solutions of SDE’s via Malliavin calculus

In this paper we develop a new method for the construction of strong solutions of stochastic equations with discontinuous coefficients. We illustrate this approach by studying stochastic differential equations driven by the Wiener process. Using Malliavin calculus we derive the result of A.K. Zvonkin (1974) [31] for bounded and measurable drift coefficients as a special case of our analysis of SDE’s. Moreover, our approach yields the important insight that the solutions obtained by Zvonkin are even Malliavin differentiable. The latter indicates that the “nature” of strong solutions of SDE’s is tightly linked to the property of Malliavin differentiability.We also stress that our method does not involve a pathwise uniqueness argument but provides a direct construction of strong solutions. © 2009 Elsevier Inc. All rights reserved