Research project is (co) funded by the Slovenian Research Agency.

**UL Member:** Faculty of Mathematics and Physics

**Code:** N1-0174

**Project:** First-passage times of one-sided Markov processes

**Period:** 1. 10. 2021 - 30. 9. 2022

**Range per year:** 1 FTE **category:** B

**Head:** Matija Vidmar

**Research activity**: Natural sciences and mathematics

**Research Organisations, researchers, citations for bibliographic records**

**Project description:**

A numerical quantity may evolve in time in a random way; we then describe it mathematically as a stochastic process. A basic, application-wise relevant - but usually very challenging - problem, is to determine the probabilities concerning the first time that such a process passes below [or above, but it is analogous] a given level. This is true even when we (as we here do) restrict attention to processes whose future stochastic evolution depends on the past only through the present (Markov processes) and which do not skip any level when attaining new minima [or maxima, but again it is analogous] (are downwards skip-free). Nevertheless, it emerges that there are some standard (well-known) families of downwards skip-free Markov processes, for which certain canonical probabilities (namely, the so-called Laplace transforms) associated with the mentioned first passage times downwards are expressed in a particularly "appealing" (this can be made technically precise) and tractable form. The main goal of this project is to (try and) identify precisely the subclass of downwards skip-free Markov processes which attain this "appealing" form, thereby bringing the hitherto known examples under a common umbrella. This should also yield: (a) further insight into the nature/properties of this type of processes; (b) the identification of new representatives of this subclass for which said form is as tractable as can be.