In probability theory, a stable process is a type of stochastic process. It includes stochastic processes whose associated probability distributions are stable distributions.[1]
Examples of stable processes include the Wiener process, or Brownian motion, whose associated probability distribution is the normal distribution. They also include the Cauchy process. For the symmetric Cauchy process, the associated probability distribution is the Cauchy distribution.[1]
The degenerate case, where there is no random element, i.e., X(t) = mt, where m is a constant, is also a stable process.[1]
References
Itô, K. (2006). Essentials of Stochastic Processes. American Mathematical Society. pp. 50–55. ISBN 9780821838983.
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
Both
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
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 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
Disciplines
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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