Additive identity

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. 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} is a Markov additive process with continuous time parameter t if

1. {(X(t),J(t)) : t ≥ 0} is a Markov process
2. the conditional distribution of (X(t + s) − X(t),J(s + t)) given (X(s),J(s)) depends only on J(s).

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

$\mathbb{𝔼}[f(X_{t+s}-X_t)g(J_{t+s})|{\mathcal{ℱ}}_t]={\mathbb{𝔼}}_{J_t,0}[f(X_s)g(J_s)]$.

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.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.