Perturbed Brownian motion reflected at a time-dependent boundary

Source abstract

Let $B$ be a standard Brownian motion, $x\ge 0,\, ν 0$, 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 $ν<1/2$ is a consequence of the Skorokhod problem in an orthant proved by [Williams 1995], while the case $ν\ge 1/2$ 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 $α\in(0,1/2)$, we also construct an increasing $α$-Hölder boundary for which no continuous adapted solution starting from zero exists for any $ν<1$.