On Boundary Crossing Probabilities for Diffusion Processes

In this paper we establish a relationship between the asymptotic form of conditional boundary crossing probabilities and first passage time densities for diffusion processes. We show that, under the assumption that the conditional probability that our diffusion (Xs, s ≥ 0) doesn’t cross an upper boundary g(·) prior to time t given that Xt = z behaves as (a+ o(1))(g(t)− z) as z ↑ g(t), there exists an expression for the first passage time density of g(·) at time t in terms of the coefficient a of the leading asymptotic term and the transition density of the diffusion process (Xs). This assumption is shown to hold true under mild conditions. We also derive a relationship between first passage time densities for diffusions and for their corresponding diffusion bridges. Finally, we prove that the probability of not crossing the boundary g(·) on the fixed time interval [0, T ] is a (Gâteaux) differentiable function of g(·) and give an explicit representation of the derivative.