Boundary crossing problems for one-dimensional Markov processes to moving boundaries

The project is concerned with boundary crossing problems (BCPs) for various Markov processes. The goal is to find explicit and closed form solutions for random times such as first or last passage time distribution of stochastic processes hitting moving boundaries. Investigating such problems are both of practical and theoretical importance. Such problems arise in many fields of sciences such as mathematical physics, mathematical finance, neurology & etc. In the case of the Brownian motion, this is a classical problem, and explicit solutions can be derived for simple boundaries such as linear, square root or quadratic. The method of images enables us to derive such results for a more complicated set of boundaries and the goal is to extend such methods to other continuous or jump Markov processes (such as Levy processes) and find new family of curves such that explicit results can be obtained.

We already published a manuscript titled "Boundary crossing problems and functional transformations for Ornstein-Uhlenbeck processes", were we investigated a two-parameter family of functional transformations and showed its connection to the first passage time (FPT) of Ornstein-Uhlenbeck(OU) type processes to time varying thresholds. Such a hitting time problem is of great interest, as the OU process has been used in many applications to model objects such as interest rates in finance or the evolution of the neuronal membrane voltages in neuroscience. The abstract and the paper itself can be found here (https://doi.org/10.48550/arXiv.2210.01658). Currently, we are interested in BCPs for jump process such as Spectrally negative Levy processes (a Levy process with no positive jumps) or generalized OU processes where instead of having a Brownian motion driving the process, we have a spectrally negative Levy process.