Generalized parking problems for levy processes

Abstract

Let X(t), 0 ≤ t < ∞ denote a real-valued process with independent and stationary increments. We assume that X has paths which are right-continuous and have left limits and that 0 < EX(1) < ∞. Let gdenote a nonnegative convex function on the real linc which assumes a unique minimum at some point b. We treat the problem of minimizing Eg(X(T)) over all stopping times T of X. We prove the existence of threshold a* ≤ b such that it is optimal to stop as soon as X(t) > a*. Under suitable conditions on g the threshold a* can be characterized in terms of the size of the jump of X over an infinite barrier. The optimal solution can also be characterized as an infinitesimal look ahead stopping rule. We present an application of our results to tests of power one.