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

Let t → h ( t ) be a smooth function on ℝ + , and B = { B s ; s ≥ 0} a standard Brownian motion. In this paper we derive expressions for the distributions of the variables T h : = inf { S ; B s = h ( s )} and λ t h : = sup { s ≦ t; B s = 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.