Final Distribution of a Diffusion Process with Final Stop

Abstract

A one-dimensional diffusion process is considered. The characteristic operator of this process is assumed to be a linear differential operator of the second order with negative coefficient at the term with zero derivative. Such an operator determines the measure of a Markov diffusion process with break (the first interpretation), and also the measure of a semi-Markov diffusion process with final stop (the second interpretation). Under the second interpretation, the existence of the limit of the process at infinity (the final point) is characterized. This limit exists on any interval almost surely with respect to the conditional measure generated by the condition that the process never leaves this interval. The distribution of the final point expressed in terms of two fundamental solutions of the corresponding ordinary differential equation, and also the distribution of the instant final stop are derived. A homogeneous process is considered as an example.