Strong solutions of some one-dimensional SDEs with random and unbounded drifts

In this paper, we are interested in the following one dimensional forward stochastic differential equation (SDE) $$d X_{t}=b(t,X_{t},\omega)d t +\sigma d B_{t},\quad 0\leq t\leq T,\quad X_{0}=\,x\in \mathbb{R},$$ where the driving noise $B_{t}$ is a $d$-dimensional Brownian motion. The drift coefficient $b:[0,T] \times\Omega\times \mathbb{R}\longrightarrow \mathbb{R}$ is Borel measurable and can be decomposed into a deterministic and a random part, i.e., $b(t,x,\omega) = b_1(t,x) + b_2(t,x,\omega)$. Assuming that $b_1$ is of spacial linear growth and $b_2$ satisfies some integrability conditions, we obtain the existence and uniqueness of a strong solution. The method we use is purely probabilitic and relies on Malliavin calculus. As byproducts, we obtain Malliavin differentiability of the solutions, provide an explicit representation for the Malliavin derivative and prove existence of weighted Sobolev differentiable flows.