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Fudan University（复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究受时间依赖边界反射的扰动布朗运动，在边界满足条件(PB)时证明强存在性和轨道唯一性，并构造无解的反例边界。
AI 中文摘要
设 $B$ 为标准布朗运动，$x\ge 0,\\, \nu<1$，且 $b:[0,\infty)\to\mathbb R$ 是从 0 开始的局部有界变差的连续函数。我们通过建立方程 \\[ W_t=(1-\nu)x+B_t+\nu M_t(W)+\frac12 L_t^0(W-b), \qquad W_t\ge b(t), \\] 的强存在性和轨道唯一性，定义了在边界 $b$ 处反射的扰动布朗运动，其中 $M_t(W):=\sup_{0\le s\le t} W_s$，过程 $L^0(W-b)$ 是过程 $W-b$ 在 0 处的半鞅局部时。我们在边界 $b$ 满足条件 (PB) 下给出了一个正面结果：对每个 $T>0$，当 $h\downarrow 0$ 时，向上增量 $\sup_{0\le s<t\le T,\\,t-s\le h}(b(t)-b(s))^+ = o(\sqrt{h})$。证明分为两种情形：$\nu<1/2$ 的情形是 [Williams 1995] 证明的象限中 Skorokhod 问题的推论，而 $\nu\ge 1/2$ 的情形结合了确定性比较估计和关于完成的往返次数的对数上界，遵循 [Chaumont 和 Doney 1999] 的策略。对于 $\alpha\in(0,1/2)$，我们还构造了一个递增的 $\alpha$-Hölder 边界，使得对于任何 $\nu<1$，从零开始的连续适应解都不存在。
英文摘要
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 the equation \[ W_t=(1-ν)x+B_t+νM_t(W)+\frac12 L_t^0(W-b), \qquad W_t\ge b(t), \] 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 $ν<1/2$ 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 $ν<1$.
Comments19 pages
