On boundary crossing probabilities for diffusion processes

The paper deals with curvilinear boundary crossing probabilities for time-homogeneous diffusion processes. First we establish a relationship between the asymptotic form of conditional boundary crossing probabilities and the first passage time density. Namely, let ττ be the first crossing time of a given boundary g(⋅) by our diffusion process (Xs,s≥0). Then, given that, for some a≥0a≥0, one has an asymptotic behaviour of the form P(τ>t∣Xt=z)=(a+o(1))(g(t)−z)P(τ>t∣Xt=z)=(a+o(1))(g(t)−z) as z↑g(t)z↑g(t), there exists an expression for the density of ττ at time tt in terms of the coefficient aa and the transition density of the diffusion process (Xs)(Xs). This assumption on the asymptotically linear behaviour of the conditional probability of not crossing the boundary g(⋅) by the pinned diffusion is then 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][0,T] is a Gâteaux differentiable function of g(⋅) and give an explicit representation of the derivative.