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). That is to say, the evolution of X(t) is governed by J(t) in the sense that for any f and g we require[1]
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![\mathbb E[f(X_{t%2Bs}-X_t)g(J_{t%2Bs})|\mathcal F_t] = \mathbb E_{J_t,0}[f(X_s)g(J_s)]](/2012-wikipedia_en_all_nopic_01_2012/I/cecad59ecb3557d9bc820aa450a1e6f9.png)
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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.