# Classical solutions of the Backward PIDE for Markov Modulated Marked Point Processes and Applications to CAT Bonds

## TL;DR

This paper establishes conditions for the existence and uniqueness of classical solutions to backward PIDEs in jump-diffusion models with Markov modulation, with applications to pricing CAT bonds.

## Contribution

It extends previous results to jump-diffusions with diffusion-modulated jump intensities, providing a theoretical foundation for financial and actuarial applications.

## Key findings

- Proves existence and uniqueness of classical solutions under new conditions.
- Extends Pham (1998) results to jump processes with diffusion modulation.
- Applies the theory to pricing of CAT bonds.

## Abstract

The objective of this paper is to give conditions ensuring that the backward partial integro differential equation associated with a multidimensional jump-diffusion with a pure jump component has a unique classical solution; that is the solution is continuous, twice differentiable in the diffusion component and differentiable in time. Our proof uses a probabilistic arguments and extends the results of Pham (1998) to processes with a pure jump component where the jump intensity is modulated by a diffusion process. This result is particularly useful in some applications to pricing and hedging of financial and actuarial instruments, and we provide an example to pricing of CAT bond.