Markov additive process

A Markov additive process (MAP) $\{(X(t),J(t)):t\geq 0\}$ 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.