currently I try to solve the following problem that I quiet struggle with. For my background I have encountered two Markov properties, one for an arbitrary Markov Process $(\Omega, \mathcal{F}, P)$ and another Markov property in the "canonical setting" on the pathspace $(S^{[0,\infty)}$. I recall here the following Definitions/Propositions. Here is the setup regarding definition
- $\tilde{\Omega} = \{f : [0,\infty) \to S\}$
- $Y$ is the coordinate process i.e. $(t \mapsto Y_t(\omega)=\omega(t))$
- $\mathcal{Y}^0_t = \sigma(Y_s : s \leq t)$
- $\mathcal{F}:=\mathcal{Y}^0_\infty = \sigma(Y_s : s \geq 0) =:\mathcal{S}^{[0,\infty)}$
- $\nu_t$ : $\tilde{\Omega} \to \tilde{\Omega}$ is the shift operator.
Markov Property for canonical setting: $Y=(Y_t)_{t \geq 0}$ has the canonical Markov property if for any $t \geq 0$ and any $\mathcal{Y}^{0}_{\infty}$-measurable random variable $Z \geq 0$ on $S^{[0,\infty)}$, we have that $$ \mathbf{E}_{\nu}[Z \circ \nu_{t} \mid \mathcal{Y}^0_t] = \mathbf{E}_{Y_t}[Z]:=\mathbf{E}_x[Z] \big\vert_{x=Y_t} \quad \mathbf{P}_{\nu} \text{ a.s.} $$
Markov Property for the usuall setting: Let $X$ be an abstract Markov process on some $(\Omega, \mathcal{F},P)$ then the Markov property takes the form $$ \mathbf{E}[\hat{Z}_t \mid \mathcal{F}^0_t] = \mathbf{E}[\hat{Z}_t \mid \sigma(X_t)]. \quad P \text{ a.s.}$$ for any $\hat{\mathcal{F}}^0_t=\sigma(X_u : u \geq t)$-measurable random variable $\hat{Z}_t \geq 0$.
My thoughts on this. Actually I believe this should be more or less chasing definitions, however I yet don`t see it. I guess the easier direction would be the if part. Here I guess that one uses that the Law of $P \circ X$ is equal to $\mathbf{P}_{\nu} \circ Y^{-1}$ (where the latter one is the concatonation of the initial distribution and the coordinate process), however I don't know how I can relate the random variables $Z$ and $\hat{Z}$ with one another, to solve this problem?
Any hints are welcome and thanks in advance!