The Kolmogorov forward equation for a distributed model of regime-switching diffusions

TL;DR

This paper derives an integro-differential equation for the densities of regime-switching diffusions with continuous state distributions, providing algorithms for solutions and methods to approximate discrete models with continuous ones.

Contribution

It introduces a novel integro-differential equation framework for regime-switching diffusions with continuous states and offers constructive solution algorithms and approximation techniques.

Findings

Existence of a constructive algorithm for solving the Cauchy problem.

Explicit solutions for certain initial distributions.

Method to approximate discrete state models by continuous models.

Abstract

For the regime-switching diffusion process with and without advection term we propose an integro-differential equation describing the densities of states continuously distributed over a segment. We demonstrate that there exists a constructive algorithm for solving the Cauchy problem. We then show that for some initial distributions of states, the solution can be found explicitly. We also discuss how a model with a discrete number of hidden states can be approximated by a model with continuously distributed states.