Perturbed Brownian motion reflected at a time-dependent boundary
Abstract
Let B be a standard Brownian motion, x≥ 0, ν<1, and b:[0,∞)→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 Wt=(1−ν)x+Bt+νMt(W)+1/2 Lt0(W−b), Wt≥ b(t), where Mt(W):=sup0≤ s≤ t Ws and the process L0(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 sup0≤ s<t≤ T, t−s≤ h(b(t)−b(s))+ = o(√h) as h↓ 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≥ 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∈(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.
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BibTeX
@misc{wang2026perturbed,
title = {{Perturbed Brownian motion reflected at a time-dependent boundary}},
author = {Chengshi Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20491},
eprint = {2609.20491},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20491},
}
RIS
TY - UNPB
TI - Perturbed Brownian motion reflected at a time-dependent boundary
AU - Chengshi Wang
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20491
UR - https://arxiv.org/abs/2609.20491
AB - Let B be a standard Brownian motion, x≥ 0, ν<1, and b:[0,∞)→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)+1/2 L_t^0(W−b), W_t≥ b(t), where M_t(W):=sup_0≤ s≤ 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≤ s<t≤ T, t−s≤ h(b(t)−b(s))^+ = o(√h) as h↓ 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≥ 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∈(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.
ER -