Time Distribution from Zero up to Beginning: of the Final Stop of Semi-Markov Diffusion Process on Interval with Unattainable Boundaries

Abstract

A one-dimensional process with continuous trajectories on a non-negative semiaxis is considered. The process has the Markov property with respect to the first exit time from any open interval (semi-Markov process). This process is called diffusion if the probability of its first exit from any symmetric neighborhood of its initial point across any boundary tends to 1/2 as the length of this neighborhood tends to zero. The time distribution from zero up to the beginning of the final interval of constancy is investigated. This distribution depends on semi-Markov transition generating functions of the process. Representation for Laplace transform of this distribution is obtained in an integral form. The integrand of this representation explains the meaning of quadratic terms of Taylor expansion of a semi-Markov transition generating function by powers of the diameter of the symmetric neighborhood of the initial point of the process. Namely, the trajectory of the process has no final interval of constancy if the coefficient of such a quadratic term is equal to zero.