An optimal stopping problem for reflecting Brownian motions

Pavel V. Gapeev and Goran Peskir

Stochastic Processes and their Applications, 2026, vol. 196, issue C

Abstract: We study the optimal stopping problemsupτE(|Bτ1|∨|Bτ2|−cτ)where (B1, B2) is a standard two-dimensional Brownian motion and c > 0 is a given and fixed constant. We first show that this problem is equivalent to the one where |Bτ1|∨|Bτ2| is replaced by |Bτ1|+|Bτ2|. Solving the latter problem we find a closed formula for the value function expressed in terms of the optimal stopping boundary which in turn is shown to be a unique solution to a nonlinear Fredholm integral equation. A key argument in the existence proof is played by a pointwise maximisation of the expression obtained by Wald-type identities. This provides tight bounds on the optimal stopping boundary describing its asymptotic behaviour for large coordinate values of (B1, B2). The solution found is applied to find the best constants in the inequalities which bound E(|Bτ1|∨|Bτ2|) or E(|Bτ1|+|Bτ2|) from above by a constant multiple of E(τ) for any stopping time τ of (B1, B2).

Keywords: Reflecting Brownian motion (Wiener process); Optimal stopping; Wald’s identity for Brownian motion; Elliptic partial differential equation; Free-boundary problem; Smooth fit; Nonlinear Fredholm integral equation; The change-of-variable formula with local time on curves/surfaces (search for similar items in EconPapers)
Date: 2026
References: Add references at CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414926000347
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:196:y:2026:i:c:s0304414926000347

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2026.104902

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().