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In applied probability, a Markov additive process (MAP) is a bivariate Markov process where the future states depends only on one of the variables.[1]

Finite or countable state space for J(t)

The process \( {\displaystyle \{(X(t),J(t)):t\geq 0\}} \) is a Markov additive process with continuous time parameter t if[1]

\( {\displaystyle \{(X(t),J(t));t\geq 0\}} \) is a Markov process
the conditional distribution of \( {\displaystyle (X(t+s)-X(t),J(t+s))} \) given ( \( {\displaystyle (X(t),J(t))} \) 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[2]

\( {\displaystyle \mathbb {E} [f(X_{t+s}-X_{t})g(J_{t+s})|{\mathcal {F}}_{t}]=\mathbb {E} _{J_{t},0}[f(X_{s})g(J_{s})]}. \)


A fluid queue is a Markov additive process where J(t) is a continuous-time Markov chain .


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.


Magiera, R. (1998). "Optimal Sequential Estimation for Markov-Additive Processes". Advances in Stochastic Models for Reliability, Quality and Safety. pp. 167–181. doi:10.1007/978-1-4612-2234-7_12. ISBN 978-1-4612-7466-7.

Asmussen, S. R. (2003). "Markov Additive Models". Applied Probability and Queues. Stochastic Modelling and Applied Probability. 51. pp. 302–339. doi:10.1007/0-387-21525-5_11. ISBN 978-0-387-00211-8.


Stochastic processes
Discrete time

Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk
Loop-erased Self-avoiding Biased Maximal entropy

Continuous time

Additive process Bessel process Birth–death process
pure birth Brownian motion
Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process
Compound Non-homogeneous Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage


Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale
Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise

Fields and other

Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model
Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process
Cox Poisson Random field Random graph

Time series models

Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models

Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models

Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models

Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c


Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems

Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem


Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe


Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Karhunen–Loève_theorem Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation
Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space
Classical Abstract


Actuarial mathematics Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics System on Chip design Stochastic analysis Time series analysis Machine learning

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