I'm trying to solve the first question of the exercise 3.19 from the Daniel Revuz, Continuous Martingale and Brownian motion book (strong Markov property section). The text is the following.
Let $X$ be a Feller process, $T$ a finite stopping time. Prove that any $\mathcal{F}_{\infty}$-measurable and positive r.v. $Z$ may be written as $\phi(\omega, \theta_T(\omega))$ where $\phi$ is $\mathcal{F}_T \otimes \mathcal{F}_\infty$-measurable. Then $$ $$ \begin{equation} \tag{1}\label{eq:id} E_\nu [Z | \mathcal{F}_T](\omega) = \int \phi(\omega,\omega') P_{X_T(\omega)}(\text{d}\omega') \quad P_\nu-\text{a.s.} \end{equation} where
- $X$ is the canonical process $\Omega := E^{\mathbb{R}_+} \to E$ where $E$ is the state space.
- $\mathcal{F}_\infty$ is the $\sigma$-algebra over $\Omega$.
- $(\mathcal{F}_t)_{t \geq 0} := \sigma(X_u : u \leq t)$
- $T$ is a stopping time wrt to $(\mathcal{F}_t)$
- $\theta_T$ is the shift operator over $\Omega$
I'm not really sure about how to proceed.
- About writing $Z$ as $\phi$ I suppose is something similar to the Doob–Dynkin lemma.
- Instead, for the identity \eqref{eq:id} I don't know what to do. To be honest, it confuses me a bit. And I know that I should use the strong Markov property but I cannot see how.
Any hint, just to start from somewhere, would be really appreciated.