Clock Systems in Stochastic System Theory

The open stochastic clock

Definition.Let the open stochastic clock for a probability space

(Ω, F_{Ω}, P)

and filtration

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

be the stochastic Moore machine with interface

(Ω ~ 1)

, state space

\tilde{Ω}

, output

r : \tilde{Ω} \to \tilde{Ω}

given by the identiy, and update

φ_{\tilde{Ω}} : \tilde{Ω} \times 1 \to \tilde{Ω}

as before.

A behavior of a open stochastic Moore machine $(A, B, X, r, φ)$ may now be defined to be a morphism out of this clock, which can be shown to be equivalent to:

a ${F_{i}}_{i \in N}$ -adapted sequence $a_{1}, a_{2}, \dots$ of $A$ -valued random variables
a ${F_{i}}_{i \in N}$ -adapted sequence $b_{1}, b_{2}, \dots$ of $B$ -valued random variables
a ${F_{i}}_{i \in N}$ -adapted sequence $x_{1}, x_{2}, \dots$ of $X$ -valued random variables

such that

r (x_{i} (ω)) P_{x_{i + 1}} (- ∣ F_{i}) = b_{i} (ω) = φ (x_{i}, a_{i}) .