Optimal stopping and maximal inequalities for geometric Brownian motion

Abstract

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτE(max0≤t≤τXt−cτ), whereX= (Xt)t≥0is geometric Brownian motion with drift μ and volatility σ > 0, and the supremum is taken over all stopping times forX. The payoff is shown to be finite, if and only if μ < 0. The optimal stopping time is given by τ*= inf {t> 0 |Xt=g*(max0≤t≤sXs)} wheres↦g*(s) is themaximalsolution of the (nonlinear) differential equationunder the condition 0 <g(s) <s, where Δ = 1 − 2μ / σ2andK= Δ σ2/ 2c. The estimate is establishedg*(s) ∼ ((Δ − 1) /KΔ)1 / Δs1−1/Δass→ ∞. Applying these results we prove the following maximal inequality:where τ may be any stopping time forX. This extends the well-known identityE(supt>0Xt) = 1 − (σ2/ 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.