Clock Systems in Stochastic System Theory

Morphisms out of the clock

Theorem.The morphisms

(\tilde{Ω}, φ_{\tilde{Ω}}) \to (X, φ_{X})

are in bijection with

X

-valued stochastic processes following the law

φ_{X}

Proof.We must show that the diagram

is equivalent to the stochastic process

x_{1}, x_{2}, \dots

following the law

φ_{X}

, e.g.

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

The RHS of this is the bottom left path in the diagram. The LHS is equal to the top right, because

P_{x_{i + 1}} (- ∣ F_{i}) (ω) = Δ (x_{i + 1}) \circ P_{1_{Ω}} (- ∣ F_{i}) (ω) = Δ (x) \circ φ_{\tilde{Ω}}

We are done.

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