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 [tau] be the first crossing time of a given boundary by our diffusion process . Then, given that, for some a>=0, one has an asymptotic behaviour of the form as z[short up arrow]g(t), there exists an expression for the density of [tau] at time t in terms of the coefficient a and the transition density of the diffusion process (Xs). This assumption on the asymptotically linear behaviour of the conditional probability of not crossing the boundary 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 on the fixed time interval [0,T] is a Gâteaux differentiable function of and give an explicit representation of the derivative.