Markov-Modulated Affine Processes

TL;DR

This paper introduces Markov-modulated affine processes (MMAPs), a flexible class of Markov processes that extend affine processes by incorporating exogenous Markov modulation, maintaining computational tractability and enabling advanced applications.

Contribution

The paper generalizes affine processes by allowing coefficients to depend on an exogenous Markov process with unbounded generator, proving existence, deriving characteristic functions, and exploring properties.

Findings

MMAPs have a computationally convenient characteristic function.

Existence of MMAPs is established via martingale problem approach.

MMAPs are applicable in various financial modeling scenarios.

Abstract

We study Markov-modulated affine processes (abbreviated MMAPs), a class of Markov processes that are created from affine processes by allowing some of their coefficients to be a function of an exogenous Markov process. MMAPs allow for richer models in various applications. At the same time MMAPs largely preserve the tractability of standard affine processes, as their characteristic function has a computationally convenient functional form. Our setup is a substantial generalization of earlier work, since we consider the case where the generator of the exogenous process $X$ is an unbounded operator (as is the case for diffusions or jump processes with infinite activity). We prove existence of MMAPs via a martingale problem approach, we derive the formula for their characteristic function and we study various mathematical properties of MMAPs. The paper closes with a discussion of several applications of MMAPs in finance.