On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary

Abstract

Let t → h(t) be a smooth function on ℝ+, and B = {Bs; s ≥ 0} a standard Brownian motion. In this paper we derive expressions for the distributions of the variables Th: = inf {S; Bs = h(s)} and λth: = sup {s ≦ t; Bs = h(s)}, where t> 0 is given. Our formulas contain an expected value of a Brownian functional. It is seen that this can be computed, principally, using Feynman–Kac&s formula. Further, we discuss in our framework the familiar examples with linear and square root boundaries. Moreover our approach provides in some extent explicit solutions for the second-order boundaries.