Strong Solutions of Some One-dimensional SDEs with Random and Unbounded Drifts

Abstract

In this paper, we are interested in the following forward stochastic differential equation (SDE): $d X_{t}=b(t,\omega,X_{t})d t +\sigma d B_{t},\, 0\leq t\leq T,\, X_{0}=x\in \mathbb{R}$, where the coefficient $b:[0,T] \times \mathbb{R}\times \Omega\longrightarrow \mathbb{R}$ is Borel measurable and of linear growth in the second variable and is adapted. The driving noise $B_{t}$ is a $d$-dimensional Brownian motion. We obtain the existence and uniqueness of a strong solution in the present situation where the drift is not necessarily deterministic. Let us mention that the technique using drift transform based on the solution of the associated backward Kolmogorov equation is not directly applicable here since the system is non-Markovian. The method we use instead is purely probabilistic and relies on Malliavin calculus. As a byproduct, we obtain Malliavin differentiability of the solutions and provide an explicit representation for the Malliavin derivative.