Regular Conditional Probability given a natural filtration of a stochastic process

Question

OK, this is kind of re-posting, but I think I can clarify the question more, so it's worth a shot.

Consider a real valued process $(X_t)_{t \leq T}$, cadlag on a probability space $(\Omega, (\mathcal{F}^\circ_t)_{t \leq T}, \mathbb{P}). \mathcal{F}^\circ_t=\sigma(X_s;s\leq t)$ is the uncompleted, natural filtration generated by $X_t$. Unfortunately $X_t$ neither has independent increments, nor is it markov. Since $\Omega$ is a Polish space, $\mathcal{F}^\circ_T$ and also $\mathcal{F}^\circ_t$ are countably generated, so we know, there exists a regular version of the conditional probability of $\mathbb{P}$ for any fixed $t$ for $\mathbb{P}$-a.a. $\omega$, i.e. for fixed $t$, $\mathbb{P}(\cdot|\mathcal{F}_t)(\omega)$ is a prob. measure f.a.a. $\omega$.

Hence we know, that for all $t\in [0,T]\cup \mathbb{Q}$, we find a regular conditional probability f.a.a. $\omega$, depending on $t$. In words: Given almost any path of the process up to time $t$, we can deduce the probablity of events, taking that information into account.

On the remaining $\omega$'s, define some meaningless measure, so we have a measure $\forall \omega$. How can I extend this to all $t$ in a reasonable way? Reasonable means: There is one Null set $N$, so that $\forall t$ $\mathbb{P}(\cdot|\mathcal{F}_t)(\omega)$, $\omega\in N^c$, is a measure Anybody seen anything like this?

I read something like this only for Markov and Feller processes using infinitesimal generators, but this cannot be carried over one to one, because we do not have a transition semigroup.

Maybe I have a deep misunderstanding here. Grateful for any objections, hints and comments.

Answer 1

Let's assume that we are working with the canonical probability space $\Omega = D(\mathbb R)$ of càdlàg functions, and $\mathbb P$ is the law of the process. I would doubt that there is a satisfactory answer at the level of maximal generality you've stated. At the very least, the measure $\mathbb P$ should be Radon. There are extremely general results on the existences of RCPs for Radon measures (cf. Leão, Fragoso and Ruffino, Regular conditional probability, disintegration of probability and Radon spaces).

The RCP is a measure-valued function $P : [0,T] \times \Omega \to \mathcal M(\Omega)$ such that for $\mathbb P$-almost every $\omega$, the measure $P(t,\omega, \cdot)$ is a version of $\mathbb P(\cdot|\mathcal F_t)$. Do you want the function $(t, \omega) \mapsto P(t,\omega,\cdot)$ to simply exist and be measurable? If so, this can be done in the wide generality stated above; see Leão et al.

Recently, I have needed more regularity properties for RCPs, namely, continuity. Consider the space $\mathcal M(\Omega)$ of Radon measures on $\Omega$ equipped with the topology of weak convergence of measures. We say that the RCP is a continuous disintegration (or continuous RCP) when it satisfies the following property: $$\mbox{if $\omega_n \to \omega$, then the measures $P(t,\omega_n,\cdot)$ converge weakly to $P(t,\omega,\cdot)$.}$$

If the law is Gaussian, then my preprint Continuous Disintegrations of Gaussian Processes gives a necessary and sufficient condition for the law $\mathbb P$ to have a continuous disintegration. I haven't thought about this in the case of càdlàg functions, but I'm pretty sure that this will extend easily. Note that this is just for fixed $t$.

To show that the map $(t, \omega) \mapsto P(t,\omega,\cdot)$ jointly continuous, a little more work is needed. As part of a larger project, Janek Wehr and I have general results in this direction for stationary, Gaussian processes. If this is what you need, I'm happy to discuss this with you further.

Open Question: If the law $\mathbb P$ is not Gaussian but at least is log-Sobolev, then all the same results should hold. This is because log-Sobolev measures satisfy very strong concentration-of-measure properties. I have some ideas how to do this, but I haven't worked out the details because I've been busy with other projects. If anybody is interested in collaborating on extending this work to the log-Sobolev case, please contact me.