L\'evy processes with partially stochastic resetting

TL;DR

This paper analyzes a Le9vy process with stochastic resetting proportional to its current position, deriving new scale functions and solving exit problems using integral equations and resolvent series.

Contribution

It introduces a novel family of scale functions for Le9vy processes with proportional stochastic resetting, expanding the analytical tools for such processes.

Findings

Derived new scale functions for the process

Solved exit problems using integral equations

Expressed the process as an SDE with existence and uniqueness

Abstract

In this paper, we solve exit problems for a L\'evy process that resets proportionally to its current position at independent Poisson epochs times. This resetting causes an additional (proportional to its current level) downward (upward) jump when the current position of the process is on the positive (negative) domain. Such a process can be expressed as an SDE, whose existence and uniqueness it discussed. All identities are given in terms of new family of scale functions. To obtain the new family of scale functions, we reduce the problem of the LT of the exit times into integral equations that are solve in terms of resolvent series.