Continuous stochastic process

In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter. Continuity is a nice property for (the sample paths of) a process to have, since it implies that they are well-behaved in some sense, and, therefore, much easier to analyze. It is implicit here that the index of the stochastic process is a continuous variable. Some authors^[1] define a "continuous (stochastic) process" as only requiring that the index variable be continuous, without continuity of sample paths: in another terminology, this would be a continuous-time stochastic process, in parallel to a "discrete-time process". Given the possible confusion, caution is needed.^[1]

Continuity almost surely

Let (Ω, Σ, P) be a probability space, let T be some interval of time, and let X : T × Ω → S be a stochastic process. For simplicity, the rest of this article will take the state space S to be the real line R, but the definitions go through mutatis mutandis if S is Rⁿ, a normed vector space, or even a general metric space.

Given a time t ∈ T, X is said to be continuous with probability one at t if

\mathbf {P} \left(\left\{\omega \in \Omega \left|\lim _{s\to t}{\big |}X_{s}(\omega )-X_{t}(\omega ){\big |}=0\right.\right\}\right)=1.

Mean-square continuity

Given a time t ∈ T, X is said to be continuous in mean-square at t if E[|X_t|²] < +∞ and

\lim _{s\to t}\mathbf {E} \left[{\big |}X_{s}-X_{t}{\big |}^{2}\right]=0.

Continuity in probability

Given a time t ∈ T, X is said to be continuous in probability at t if, for all ε > 0,

\lim _{s\to t}\mathbf {P} \left(\left\{\omega \in \Omega \left|{\big |}X_{s}(\omega )-X_{t}(\omega ){\big |}\geq \varepsilon \right.\right\}\right)=0.

or equivalently

\lim _{s\to t}\mathbf {P} \left(\left\{\omega \in \Omega \left|{\big |}X_{s}(\omega )-X_{t}(\omega ){\big |}<\varepsilon \right.\right\}\right)=1.

Also equivalent, X is continuous in probability at time t if

\lim _{s\to t}\mathbf {E} \left[{\frac {{\big |}X_{s}-X_{t}{\big |}}{1+{\big |}X_{s}-X_{t}{\big |}}}\right]=0.

Continuity in distribution

Given a time t ∈ T, X is said to be continuous in distribution at t if

\lim _{s\to t}F_{s}(x)=F_{t}(x)

for all points x at which F_t is continuous, where F_t denotes the cumulative distribution function of the random variable X_t.

Sample continuity

X is said to be sample continuous if X_t(ω) is continuous in t for P-almost all ω ∈ Ω. Sample continuity is the appropriate notion of continuity for processes such as Itō diffusions.

Feller continuity

X is said to be a Feller-continuous process if, for any fixed t ∈ T and any bounded, continuous and Σ-measurable function g : S → R, E^x[g(X_t)] depends continuously upon x. Here x denotes the initial state of the process X, and E^x denotes expectation conditional upon the event that X starts at x.

Relationships

The relationships between the various types of continuity of stochastic processes are akin to the relationships between the various types of convergence of random variables. In particular:

continuity with probability one implies continuity in probability;
continuity in mean-square implies continuity in probability;
continuity with probability one neither implies, nor is implied by, continuity in mean-square;
continuity in probability implies, but is not implied by, continuity in distribution.

It is tempting to confuse continuity with probability one with sample continuity. Continuity with probability one at time t means that P(A_t) = 0, where the event A_t is given by

A_{t}=\left\{\omega \in \Omega \left|\lim _{s\to t}{\big |}X_{s}(\omega )-X_{t}(\omega ){\big |}\neq 0\right.\right\},

and it is perfectly feasible to check whether or not this holds for each t ∈ T. Sample continuity, on the other hand, requires that P(A) = 0, where

A=\bigcup _{t\in T}A_{t}.

A is an uncountable union of events, so it may not actually be an event itself, so P(A) may be undefined! Even worse, even if A is an event, P(A) can be strictly positive even if P(A_t) = 0 for every t ∈ T. This is the case, for example, with the telegraph 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 Biased Maximal entropy
Continuous time	Additive process Airy process Bessel process Birth–death process pure birth Brownian motion Bridge Dyson 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 Hawkes 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 Quasimartingale 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 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 Determinantal 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	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White Korn-Kreer-Lenssen 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 Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
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 Stochastic analysis Time series analysis Machine learning
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