Let $(\Omega, F, (F_t)_{t\geq 0}, \mathbb{P})$ assume that $(X_t)_{t\leq T} $ is some cadlag, real valued stochastic process, not too bad: say something like a Brownian Motion and some Poisson finite activity jump process. The filtration is generated by $X_t$ in the usual way $F_t = \bigvee_{s\leq t} F_s$. We take the right continuous version of this filtration. If we include all Null sets of $F_T$ to get the usual assumptions, I am worried the filtration is then too large for my purpose (read on).

To have all the nice properties needed for Stochastic Analysis anyway and have a sufficiently small filtration at the same time, it should be possible to use the "natural" augmentation instead: sets which are contained in a countable union $(B_n)_{n\geq0}$ of sets of probability zero, such that $B_n \in F_{t_n}$ for all $n \geq 0$, where $t_n$ is some unbounded sequence in $\mathbb{R}^+$. (see Bichteler-Stochastic Integration with Jumps or Najnudel&Nikeghbali-A new kind of augmentation of filtrations)

Now, I want to know the probability of $X_T$ crossing a certain threshold $K$. Assume that we have a density, so for $t=0$ we have simply $\int_{[K,\infty]} f d\lambda$. I then applied some tricks to compute the integral. However, I would like to extend this for conditional expectations. First thought: Regular conditional probability. Assume that we can find one $\forall t>0$. We then have a measure for each level $x$ of $X_t$ for a.a. $x$. On the remaining x in the Null $N(t)$ set, one just defines some probability measure. Assuming then that each of these measures is abs. cont. w.r.t to the Lebesgue measure, I can apply the same trick as before. The Null sets $N(t)$ vary with t, so they are uncountable. So to apply the result using the density, I think I need one Null set $N$, independent of $t$, so I can compute the integral $\forall t$ for a.a. paths. Of course I have such an $N$ for a dense subset of $[0,T]$.

Let $P_{t,x}(\cdot)$ be the regular conditional probability in (t,x). Is it possible to establish weak*-continuity in $t$? Has anybody seen anything the like? I checked the internet for papers and books like Rao etc., though very elaborate, I didn't find a hint. So I am grateful for tips, or rectification. Maybe I come up with sth on the weekend...

Cheers