An optimal stopping problem for reflecting Brownian motions

We study the optimal stopping problem supτE(|B τ 1|∨|B τ 2|−cτ)where (B 1, B 2) 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 |B τ 1|∨|B τ 2| is replaced by |B τ 1|+|B τ 2|. 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 (B 1, B 2). The solution found is applied to find the best constants in the inequalities which bound E(|B τ 1|∨|B τ 2|) or E(|B τ 1|+|B τ 2|) from above by a constant multiple of E(τ) for any stopping time τ of (B 1, B 2).