GENERAL AN ELEMENTARY DERIVATION OF THE HITTING TIME DISTRIBUTION OF BROWNIAN MOTION WITH DRIFT

Let B-B, s >= 0 be a standard Brownian motion. For given constants mu and b not equal 0, denote by T the time at which the motion B-S + mu(s), s >= 0 first hits the barrier b. The derivation of the Laplace transform of T is standard fare in courses on continuous time stochastic processes. Typically, the method is to find first the Laplace transform of T when mu = 0 and then to deduce the general result from this special case. In this note an elementary proof is given of the general result. A well-known explicit expression for the distribution function of T is found by using just one standard Laplace transform and some elementary calculus. The proofs used here could be useful in introductory postgraduate courses on stochastic processes and continuous time finance.