Optimal local first exit time

A random process and the corresponding class of so called local first exit times are considered. For a special functional depending on Markov times the problem to find the optimal one is investigated. A description of the class is obtained. For diffusion Markov processes the folowing alternative is proved: either the global first exit time is optimal (trivial case), or in the given class there are no optimal Markov times. For a non-Markov piece-wise increasing process a non-trivial example of the local first exit time is constructed. An application of the problem to insurance is discussed. Bibl. – 7 titles.