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An optimal stopping problem for reflecting Brownian motions

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  • Gapeev, Pavel V.
  • Peskir, Goran

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).

Suggested Citation

  • Gapeev, Pavel V. & Peskir, Goran, 2026. "An optimal stopping problem for reflecting Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 196(C).
    • Handle: RePEc:eee:spapps:v:196:y:2026:i:c:s0304414926000347
    DOI: 10.1016/j.spa.2026.104902
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