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Stochastic Processes (Definition)

Mathematical objects usually defined as families of random variables, representing systems that change randomly over time.
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The statement of the theorem

Let (Ω,F,P)(\Omega, \mathcal{F}, P) be a complete probability space, and let SS be a measurable state space (e.g., S=RdS = \mathbb{R}^d). A stochastic process XX indexed by a set TT (the index set, typically T=[0,∞)T = [0, \infty) or T=ZT = \mathbb{Z}) is formally defined as a collection of random variables X={Xt}t∈TX = \{X_t \}_{t \in T} such that:\n\n1. **Measurability:** For every t∈Tt \in T, the random variable Xt:Ω→SX_t: \Omega \to S is F\mathcal{F}-measurable.\n2. **Joint Measurability:** The mapping X:Ω×T→SX: \Omega \times T \to S defined by X(ω,t)=Xt(ω)X(\omega, t) = X_t(\omega) must be measurable with respect to the product σ\sigma-algebra F⊗B(T)\mathcal{F} \otimes \mathcal{B}(T), where B(T)\mathcal{B}(T) is the Borel σ\sigma-algebra on TT. \n\nAlternatively, the process XX induces a filtration FT=σ{Xt}t∈T\mathcal{F}_T = \sigma\{X_t \}_{t \in T} on F\mathcal{F}, which represents the information available up to time TT. If TT is continuous, the process is often defined as a measurable function X:Ω×T→SX: \Omega \times T \to S such that for any measurable set A⊂SA \subset S, the mapping (ω,t)↦1A(Xt(ω))(\omega, t) \mapsto \mathbf{1}_A(X_t(\omega)) is measurable with respect to F⊗B(T)\mathcal{F} \otimes \mathcal{B}(T).