- Source: 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.
Definition
= Finite or countable state space for J(t)
=The process
{
(
X
(
t
)
,
J
(
t
)
)
:
t
≥
0
}
{\displaystyle \{(X(t),J(t)):t\geq 0\}}
is a Markov additive process with continuous time parameter t if
{
(
X
(
t
)
,
J
(
t
)
)
;
t
≥
0
}
{\displaystyle \{(X(t),J(t));t\geq 0\}}
is a Markov process
the conditional distribution of
(
X
(
t
+
s
)
−
X
(
t
)
,
J
(
t
+
s
)
)
{\displaystyle (X(t+s)-X(t),J(t+s))}
given
(
X
(
t
)
,
J
(
t
)
)
{\displaystyle (X(t),J(t))}
depends only on
J
(
t
)
{\displaystyle 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
E
[
f
(
X
t
+
s
−
X
t
)
g
(
J
t
+
s
)
|
F
t
]
=
E
J
t
,
0
[
f
(
X
s
)
g
(
J
s
)
]
{\displaystyle \mathbb {E} [f(X_{t+s}-X_{t})g(J_{t+s})|{\mathcal {F}}_{t}]=\mathbb {E} _{J_{t},0}[f(X_{s})g(J_{s})]}
.
Example
A fluid queue is a Markov additive process where J(t) is a continuous-time Markov chain.
Applications
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.