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 {(X(t),J(t)) : t ≥ 0} is a Markov additive process with continuous time parameter t if[1]

1. {(X(t),J(t)) : t ≥ 0} is a Markov process
2. the conditional distribution of (X(t + s) − X(t),J(s + t)) given (X(t),J(t)) depends only on 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]

$\mathbb E[f(X_{t+s}-X_t)g(J_{t+s})|\mathcal F_t] = \mathbb E_{J_t,0}[f(X_s)g(J_s)]$.

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