conccccccxx # More information Consider this model again: $$x_{k+1}=f(x_{k},u_{k},w_{k})$$

^statespace

where $x_{t}\in\mathbb{X}$, $u_{t}\in\mathbb{U}$, $w_{t}\in\mathbb{W}$, $f$ a measurable function, $\mathbb{X},\mathbb{U},\mathbb{W}$ are standard Borel. We assume all random variables live in some probability space $(\Omega,\mathcal{F},P)$. The collection, $\{ (x_{k},u_{k}) \}$, satisfying also satisfies

$$\begin{align*} P(x_{k+1}\in B|x_{[0,k]}=a_{[0,t]},u_{[0,k]}=b_{[0,t]})&= P(x_{k+1}\in B|x_{k}=a_{k},u_{k}=b_{k})\\ &=:\mathcal{T}(B\mid a_{t},b_{t}) \end{align*}$$ ^property

$\forall B\in\mathcal{B}(\mathbb{R}),k\in\mathbb{Z}_{+}$, where $\mathcal{T}(\cdot\mid x,u)$ is a Stochastic Kernel s.t. $\mathcal{T}:\mathbb{X}\times \mathbb{U}\to \mathbb{X}$ so that: >- For every $B\in\mathcal{B}(\mathbb{R})$, $\mathcal{T}(B\mid \cdot,\cdot)$ is a measurable function on $\mathbb{X}\times \mathbb{U}$ and; >- For every fixed $(a,b)\in\mathbb{X}\times \mathbb{U}$, $\mathcal{T}(\cdot\mid x,u)$ is a probability measure on $(\mathbb{X},\mathcal{B}(\mathbb{X}))$.

That is, all Stochastic Processes that satisfy , admit a Stochastic Realization in the form of almost surely.