# Characterizing filtrations generated by a stopping time

## Setup

Let $$\Omega$$ be the set of càdlàg functions $$f : [0,\infty) \to \mathbb R^d$$ equipped with the Skorokhod topology for any $$d \geq 1$$, and let $$X_t(\omega) = \omega(t)$$ for any $$\omega \in \Omega, t \geq 0$$. Then, $$\{X_t(\omega)\}_{t \geq 0}$$ is a stochastic process on $$\mathbb R^d$$ with associated natural filtration $$\{\mathcal F_{t}\}_{t \geq 0}$$ (and we let $$\mathcal F_{\infty} = \sigma(\cup_{n \geq 0} \mathcal F_n)$$). Let $$\tau$$ be an a.s. finite stopping time, and recall the stopped filtration $$\mathcal F_{\tau}$$, which contains all $$A \in \mathcal F_{\infty}$$ such that $$A \cap \{\tau \leq t\} \in \mathcal F_t$$ for all $$t \geq 0$$.

My question is concerned with a very "logical" but, to me, intractable characterization of $$\mathcal F_{\tau}$$ for a stopping time $$\tau$$ in this particular case. Let $$\tau_1 \leq \tau_2$$ be two stopping times. For any stopping time $$\sigma$$ we can define the "shift" operator $$\theta_{\sigma}:\Omega \to \Omega$$ by $$(\theta_{\sigma}(\omega))(t) = \omega(t + \sigma(\omega))$$ (i.e. just shifting time ahead by $$\sigma$$).

## Motivation and question

Informally, what is the filtration $$\mathcal F_{\tau_2}$$? It consists of all events $$A$$ whose occurrence or lack of is completely determined by "all coordinates before $$\tau_2$$" (or determined by observing the stochastic process till time $$\tau_2$$). Now, break any such observation into two parts :

• A part based on the observations purely from $$\tau_1$$.

• A part based on the observations between $$\tau_1$$ and $$\tau_2$$.

Now, we know that events determined purely by the first category of observations are $$\mathcal F_{\tau_1}$$. On the other hand, events determined purely by the second category of observations are captured by sets of the form $$\{\omega : \theta_{\tau_1}(\omega) \in B\}$$ where $$B \subset \mathcal F_{\tau_{2} - \tau_1}$$.

Hence, the following question arises.

For $$A \in \mathcal F_{\tau_1}$$, $$B\in \mathcal F_{\tau_2 - \tau_1}$$, let $$E_{A,B} = A \cap \{\omega : \theta_{\tau_1}(\omega) \in B\}.$$ Is it true that $$\mathcal F_{\tau_2} = \sigma(\{E_{A,B} : A \in \mathcal F_{\tau_1}, B\in \mathcal F_{\tau_2 - \tau_1}\})?$$

## Ideas and known results

This question on Mathematics Stack Exchange is a corollary of my question above, so that's why I'd like an answer.

Here are some known results.

• $$\mathcal F_{\tau} = \sigma\{X_{\tau \wedge t} : t\geq 0\}$$ for any a.s. finite stopping time $$\tau$$.

• A function $$f : \Omega \to \mathbb R^d$$ is $$\mathcal F_{\tau}$$ measurable if and only if, for any $$\omega_1,\omega_2$$ such that $$\tau_1(\omega) = \tau_2(\omega)$$, we have $$f(\omega_1) = f(\omega_2)$$. In particular, if $$f$$ is $$\mathcal F_{\tau}$$ measurable, then $$f(\omega) = f(\omega^{\tau})$$ for all $$\omega \in \Omega$$, where $$\omega^{\tau}(t) = \omega(t \wedge \tau(\omega))$$.

More ideas may be found in the MSE question linked above. I'm surprised an answer isn't available in a standard text.

$$\newcommand\F{\mathcal F}$$This question does not seem to make sense, because the symbol $$\F_{\tau_2-\tau_1}$$ does not make sense in general, because $$\tau_2-\tau_1$$ will not be a stopping time in general even if $$\tau_1$$ and $$\tau_2$$ are stopping times such that $$\tau_1\le\tau_2$$.
For instance, suppose that $$X_t=1+B_t$$, where $$B_\cdot$$ is a standard Brownian motion, $$\tau_2:=\inf\{t\ge0\colon X_t=2\},\quad \tau_0:=\inf\{t\ge0\colon X_t=0\},$$ $$\tau_1:=\tau_0\wedge\tau_2=\inf\{t\ge0\colon X_t\in\{0,2\}\}\le\tau_2.$$ Then the event $$\{\tau_2-\tau_1\le1\} =\big\{\exists s\ge0\ X_s=2\ \&\ \forall u\in[0,s-1)\ X_u\notin\{0,2\}\big\}$$ is not determined by $$(X_t)_{t\in[0,1]}$$.