# Markov additive process

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[edit]

### Finite or countable state space for *J*(*t*)[edit]

The process is a Markov additive process with continuous time parameter *t* if^{[1]}

- is a Markov process
- the conditional distribution of given depends only on .

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*)[edit]

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]}

- .

## Example[edit]

A fluid queue is a Markov additive process where *J*(*t*) is a continuous-time Markov chain^{[clarification needed]}^{[example needed]}.

## Applications[edit]

This section may be confusing or unclear to readers. (April 2020) |

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[edit]

- ^
^{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. **^**Asmussen, S. R. (2003). "Markov Additive Models".*Applied Probability and Queues*. Stochastic Modelling and Applied Probability. Vol. 51. pp. 302–339. doi:10.1007/0-387-21525-5_11. ISBN 978-0-387-00211-8.