In applied probability, a Markov additive process (MAP) is a bivariate Markov process where the future states depends only on one of the variables.[1]

## Definition

### Finite or countable state space for J(t)

The process ${\displaystyle \{(X(t),J(t)):t\geq 0\}}$ is a Markov additive process with continuous time parameter t if[1]

1. ${\displaystyle \{(X(t),J(t));t\geq 0\}}$ is a Markov process
2. the conditional distribution of ${\displaystyle (X(t+s)-X(t),J(t+s))}$ given ${\displaystyle (X(t),J(t))}$ depends only on ${\displaystyle J(t)}$.

The state space of the process is R × S where X(t) takes real values and J(t) takes values in some countable set S.

### General state space for J(t)

For the case where J(t) takes a more general state space the evolution of X(t) is governed by J(t) in the sense that for any f and g we require[2]

${\displaystyle \mathbb {E} [f(X_{t+s}-X_{t})g(J_{t+s})|{\mathcal {F}}_{t}]=\mathbb {E} _{J_{t},0}[f(X_{s})g(J_{s})]}$.

## Example

A fluid queue is a Markov additive process where J(t) 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.

## Notes

1. ^ a b Magiera, R. (1998). "Optimal Sequential Estimation for Markov-Additive Processes". Advances in Stochastic Models for Reliability, Quality and Safety. pp. 167–181. doi:10.1007/978-1-4612-2234-7_12. ISBN 978-1-4612-7466-7.
2. ^ Asmussen, S. R. (2003). "Markov Additive Models". Applied Probability and Queues. Stochastic Modelling and Applied Probability. 51. pp. 302–339. doi:10.1007/0-387-21525-5_11. ISBN 978-0-387-00211-8.