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 τ be the first crossing time of a given boundary g ( ⋅ ) by our diffusion process ( X s , s ≥ 0 ) . Then, given that, for some a ≥ 0 , one has an asymptotic behaviour of the form P ( τ > t ∣ X t = z ) = ( a + o ( 1 ) ) ( g ( t ) − z ) as z ↑ g ( t ) , there exists an expression for the density of τ at time t in terms of the coefficient a and the transition density of the diffusion process ( X s ) . 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 ] is a Gâteaux differentiable function of g ( ⋅ ) and give an explicit representation of the derivative.