Optimal stopping of the maximum process: a converse to the results of Peskir

Peskir, (and also Meilijson and Ob lój) considered the following optimal stopping problem: find, for an increasing function F and a positive function λ, sup τ E [ F (Sτ ) − ∫ τ 0 λ(Bu)du ] , (1) where S is the maximum process of Brownian motion. In this article we are interested in the converse: find, for an increasing function F and a suitable function λ, sup τ E [ ∫ τ 0 λ(Bu)du − F (Sτ ) ] . In the non-degenerate cases the optimal stopping rule is of the form stop the first time that St reaches γ or Bt falls below g(St) where γ, a positive constant, and g, a negative function, are both to be chosen. The optimal function g is characterised as the solution to non-linear differential equation, which is very similar to that used by Peskir to characterise the solution to (1), however we derive this differential equation in a completely different way.