Clock Systems in Stochastic System Theory

Stochastic processes, correct definition

Definition.A filtration of a measurable space

(Ω, F_{Ω})

is a sequence

F_{1} \subseteq F_{2} \subseteq \dots \subseteq F_{Ω}

of sub-

σ

-algebras.

Definition.A sequence of random variables

x_{1}, x_{2}, \dots : Ω \to X

is adapted to a filtration

F_{1} \subseteq F_{2} \subseteq \dots \subseteq F_{Ω}

x_{i}

is measurable with respect to

F_{i}

Intuition: $F_{i}$ represents the questions we can ask at time $i$ , and $x_{i}$ must be discernable given answers to these questions.

Definition.A stochastic process

x_{1}, x_{2}, \dots : Ω \to X

adapted to

F_{1} \subseteq F_{2} \subseteq \dots \subseteq F_{Ω}

satisfies the law

φ : X \to Δ (X)

if for all

i \in N

P

-almost all

ω \in Ω

P_{x_{i + 1}} (- ∣ F_{i}) (ω) = φ (x_{i} (ω)) : Δ (X)