Perturbed Brownian motion reflected at a time-dependent boundary

RESUMO

Let $B$ be a standard Brownian motion, $x\ge 0,\, \nu<1$, 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 the equation \[ W_t=(1-\nu)x+B_t+\nu M_t(W)+\frac12 L_t^0(W-b), \qquad W_t\ge b(t), \] where $M_t(W):=\sup_{0\le s\le t} W_s$ and the process $L^0(W-b)$ is the semimartingale local time at 0 of the process $W-b$. We give a positive result under condition (PB) on the boundary $b$ : for every $T>0$, the upward increment $\sup_{0\le s<t\le T,\,t-s\le h}(b(t)-b(s))^+ = o(\sqrt{h})$ as $h\downarrow 0$. The proof splits into two regimes: the case $\nu<1/2$ is a consequence of the Skorokhod problem in an orthant proved by [Williams 1995], while the case $\nu\ge 1/2$ combines a deterministic comparison estimate and a logarithmic upper bound on the number of completed round-trips, following the strategy of [Chaumont and Doney 1999]. For $\alpha\in(0,1/2)$, we also construct an increasing $\alpha$-Hölder boundary for which no continuous adapted solution starting from zero exists for any $\nu<1$.