Perturbed Brownian motion reflected at a time-dependent boundary

Let $B$ be a standard Brownian motion, $x\ge 0,\, ν<1$, and $b:[0,\infty)\to\mathbb R$ is a continuous function locally of finite variation starting from 0. We define the perturbed Brownian motion reflected at the boundary $b$ by establishing strong existence and pathwise uniqueness of a solution to