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 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.

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

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

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

Equivalently, X is continuous in probability at time t if

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

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