Piecewise linear processes with Poisson-modulated exponential switching times

TL;DR

This paper studies a jump telegraph process with Poisson-modulated exponential switching times, analyzing its application in incomplete financial markets and the use of Esscher transform for risk pricing.

Contribution

It introduces a novel jump telegraph process with external shocks and explores the Esscher transform for pricing risks in incomplete markets.

Findings

The process models switching intensities dependent on external shocks.

The Esscher transform effectively prices switching and jump risks.

Application to incomplete financial market models.

Abstract

We consider the jump telegraph process when switching intensities depend on external shocks also accompanying with jumps. The incomplete financial market model based on this process is studied. The Esscher transform, which changes only unobservable parameters, is considered in detail. The financial market model based on this transform can price switching risks as well as jump risks of the model.