Final distribution of a~diffusion process with a~final stop

B. P. Harlamov

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 21, Tome 431 (2014), pp. 209-241

Résumé

One-dimensional diffusion process is considered. A characteristic operator of this process is assumed to be a linear differential operator of the second order with a negative coefficient in the operator's member without derivative. Such an operator determines a measure of a Markov diffusion process with a break (the first interpretation), and also that of a semi-Markov diffusion process with a final stop (the second interpretation). Under the second interpretation the existence of a limit on infinity of the process (the final point) is characterized. This limit exists on any interval almost sure with respect to a conditional measure, generated by condition that the process never leaves this interval. A distribution of the final point expressed in terms of two fundamental solutions of the corresponding ordinary differential equation, and also that of the final stop beginning instant are derived. A homogeneous process is considered as an example.