In applied probability, a Markov additive process (MAP) is a bivariate

Markov process A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...

where the future states depends only on one of the variables.

Definition

Finite or countable state space for ''J''(''t'')

The process

\

is a Markov

additive process An additive process, in probability theory, is a cadlag, Continuous stochastic process#Continuity in probability, continuous in probability stochastic process with independent increments. An additive process is the generalization of a Lévy process ...

with continuous time parameter ''t'' if #

\

is a

# the conditional distribution of

(X(t+s)-X(t), J(t+s))

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 ::

\mathbb E \mathcal F_t = \mathbb E_(X_s)g(J_s) /math>.

Example

fluid queue In queueing theory, a discipline within the mathematical theory of probability, a fluid queue (fluid model, fluid flow model or stochastic fluid model) is a mathematical model used to describe the fluid level in a reservoir subject to randomly deter ...

is a Markov additive process where ''J''(''t'') is a

continuous-time Markov chain A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of ...

Applications

Çinlar uses the unique structure of the MAP to prove that, given a

gamma process In mathematics and probability theory, a gamma process, also known as (Moran-)Gamma subordinator, is a random process with independent gamma distributed increments. Often written as \Gamma(t;\gamma,\lambda), it is a pure-jump increasing Lévy ...

with a shape parameter that is a function of

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

, the resulting lifetime is distributed according to the

Weibull distribution In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice Re ...

. 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 A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the to ...

Notes

{{probability-stub Markov processes