Markov additive process

In applied probability, a Markov additive process (MAP) {(X(t),J(t)) : t ≥ 0} is a bivariate Markov process whose transition probability measure is translation invariant in the additive component X(t).

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.