Discrete-time stochastic process

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In probability theory and statistics, a discrete-time stochastic process is a stochastic process for which the index variable takes a discrete set of values, as contrasted with a continuous-time process for which the index variable takes values in a continuous range. An alternative terminology uses discrete parameter as being more inclusive.^[1]

A more restricted class of processes are those with discrete time and discrete state space. The apparently simpler terms "discrete process" or "discontinuous process" may cause confusion with processes having continuous time and discrete state space.^[2] Given the possible confusion, caution is needed.^[2]

Examples

Examples of discrete-time stochastic processes are random walks and branching processes, for which the state space may be either continuous or discrete. Important examples of discrete time and continuous state space processes are models conventionally used in time series analysis: for example, the autoregressive, vector autoregressive, moving average, ARMA, ARIMA and ARCH models.

References

↑ Parzen, E. (1962) Stochastic Processes, Holden-Day. ISBN 0-8162-6664-6 (page 7)
↑ ^2.0 ^2.1 Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 (Entry for "continuous process")

v t e 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
Continuous time	Bessel process Birth–death process 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 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 Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process 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
Properties	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
Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe
Tools	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 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
Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning
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