PRESERVATION OF THE MARKOV PROPERTY UNDER DELAYED REFLECTION

摘要

A one-dimensional locally Markov diffusion process with positive range of values is considered. It is assumed that this process is reflected from the point 0. All the variants of reflection with preservation of the semi-Markov property are described. The reflected process is still locally Markov in open intervals, but it can lose the global Markov property. The reflection is characterized by the time α(r) which is the first exit time from the semi-interval [0,r) after the first hitting time at level 0 (for r ＞ 0). A distribution of this time is used to derive a time change transforming a process with instantaneous reflection into a process with delayed reflection. It is proved that a process that preserves its Markov property after the delayed reflection has a special distribution of the set of times at which the process has zero values during the interval [0, α(r)). For such a process, this set has exponentially distributed Lebesgue measure. Bibliography: 7 titles.