Clock Systems in Stochastic System Theory

Closed stochastic clock system

Definition.Given a probability space

(Ω, F_{Ω}, P)

, with filtration

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

, there is a closed stochastic system with state space

\tilde{Ω} = i : N \sum (Ω, F_{i})

and update function given by

φ_{\tilde{Ω}} ((i, ω)) = P_{1_{Ω}} (- ∣ F_{i}) (ω)

Proposition.Measurable functions

\tilde{Ω} \to X

are in bijection with stochastic processes adapted to

{F_{i}}_{i \in N} .

Proof.By the universal property of coproduct, a measurable function

\tilde{Ω} \to X

is equivalent to a collection

{x_{i} : Ω \to X}_{i \in N}

where

x_{i}

is measurable with respect to

F_{i}

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