A Multivariate Stochastic Hybrid Model with Switching Coefficients and Jumps: Solution and Distribution

D P Siu,G S Ladde

doi:10.1155/2011/720614

Abstract

In this work, a class of multidimensional stochastic hybrid dynamic models is studied. The system under investigation is a first‐order linear nonhomogeneous system of Itô‐Doob type stochastic differential equations with switching coefficients. The switching of the system is governed by a discrete dynamic which is monitored by a non‐homogeneous Poisson process. Closed‐form solutions of the systems are obtained. Furthermore, the major part of the work is devoted to finding closed‐form probability density functions of the solution processes of linear homogeneous and Ornstein‐Uhlenbeck type systems with jumps.

Highlights

The study of stochastic hybrid systems exhibiting both continuous and discrete dynamics has been an area of great interest over the years
Davis 1, 2 introduced a piecewisedeterministic Markov process, where transitions between discrete modes are triggered by random events and deterministic conditions for hitting the boundary, while the continuousstate process between jumps for the model is governed by a deterministic differential equation
Hu et al 4 proposed a stochastic hybrid system where the deterministic differential equations for the evolution of the continuous-state process are replaced by Ito -Doob type stochastic differential equations 5, 6

Summary

Introduction

The study of stochastic hybrid systems exhibiting both continuous and discrete dynamics has been an area of great interest over the years. Hu et al 4 proposed a stochastic hybrid system where the deterministic differential equations for the evolution of the continuous-state process are replaced by Ito -Doob type stochastic differential equations 5, 6 In this proposed model, the transitions are only triggered by hitting the boundaries. We attempt to solve two fundamental problems in the stochastic modeling of dynamic processes described by Ito -Doob type stochastic differential equations with jumps. This is an extension of the geometric Brownian motion processes 20.

Model Formulation

Probability Distribution of One-Dimensional Linear Homogeneous Models

Probability Distribution of Multivariate Linear Homogeneous Models

Probability Distribution of an Ornstein-Uhlenbeck Model with Jumps

Concluding Remarks