# Feynman-Kac Formulas for Regime-Switching Jump Diffusions and their Applications

## TL;DR

This paper derives Feynman-Kac formulas for complex regime-switching jump diffusions driven by Lévy processes, linking stochastic processes with partial integro-differential equations and demonstrating convergence to the arcsine law.

## Contribution

It introduces Feynman-Kac formulas for regime-switching jump diffusions with Lévy jumps and establishes their connection to PDEs under broad conditions.

## Key findings

- Connection between stochastic processes and PDEs established.
- Convergence of related random variables to the arcsine law shown.
- Framework applicable to initial, terminal, and boundary value problems.

## Abstract

This work develops Feynman-Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated to a general L\'evy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal, and boundary value problems are also treated. Moreover, based on weak convergence of probability measures, it is demonstrated that a sequence of random variables related to the regime-switching jump diffusion process converges in distribution to the arcsine law.