Markov additive process


In applied probability, a Markov additive process is a bivariate Markov process where the future states depends only on one of the variables.

Definition

Finite or countable state space for ''J''(''t'')

The process is a Markov additive process with continuous time parameter t if
  1. is a Markov process
  2. the conditional distribution of given depends only on.
The state space of the process is R × S where X takes real values and J takes values in some countable set S.

General state space for ''J''(''t'')

For the case where J takes a more general state space the evolution of X is governed by J in the sense that for any f and g we require

Example

A fluid queue is a Markov additive process where J is a continuous-time Markov chain.

Applications

Çinlar uses the unique structure of the MAP to prove that, given a gamma process with a shape parameter that is a function of Brownian motion, the resulting lifetime is distributed according to the Weibull distribution.
Kharoufeh presents a compact transform expression for the failure distribution for wear processes of a component degrading according to a Markovian environment inducing state-dependent continuous linear wear by using the properties of a MAP and assuming the wear process to be temporally homogeneous and that the environmental process has a finite state space.