Difference between $\mathcal F_\tau$ and $\sigma(\tau)$

Question

suppose we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with a filtration $(\mathcal{F}_n)_{n \in \mathbb N}$. A stopping time is random variable $\tau$ such that $$ \{\tau \leq n\} \in \mathcal F_n, \quad n \in \mathbb N. $$ For a given stopping time $\tau$ we can define a $\sigma$-algebra as $$ \mathcal{F}_\tau= \big\{A \in \mathcal F\mid A \cap \{\tau \leq n\} \in \mathcal F_n, \ n \in \mathbb{N} \big\}. $$

The usual interpretation is that for $n \in \mathbb{N}, \ $ $\mathcal F_n$ contains all the information up to time $n$ and $\mathcal F_\tau$ contains all the information up to time $\tau$.

My question is: what is the intuitive difference between $\mathcal F_\tau$ and $\sigma(\tau)$, i.e. the $\sigma$-algebra generated by $\tau$. For example, if $X=(X_n)_{n =\{0,\dots, N\}}$ denotes a gamblers stack and $\tau$ is a rule which the gambler uses to end the game, what is the difference between $$ \mathbb{E}[X_N\mid \mathcal F_\tau ] \quad \text{and} \quad \mathbb{E}[X_N\mid \tau ]? $$ Thank you

Answer 1

As you say, $\mathcal{F}_\tau$ contains all the information up to the stopping time, while $\sigma(\tau)$ only contains information that is relevant to the stopping.

For example, take $X_1,X_2,\ldots,X_N$ iid with continuous distribution, $\tau=\inf\{i:X_{i-1}>0,X_i>0\}\wedge N$, and $S_k=\sum_{i=1}^{k\wedge \tau}X_i$. That is, a player plays the same game until he gets a profit twice in a row. Then the total result of the game, $S_N$, actually equals $S_\tau$, and so it is $\mathcal{F}_\tau$-measurable. However, it is far from $\sigma(\tau)$-measurable, in fact knowing the value of $\tau$ doesn't even tell you the sign of $S_N$, that is, whether the player made a total profit or loss (unless of course $\tau=2$).

Answer 2

This is just a little more detailed explanation which I derived from m7e's answer. I thought about his/her answer a lot and came up whit this explanation which I personally like a lot:

• $\sigma(\tau)$ contains all the information on when $\tau$ has happened
• $\mathcal F_\tau$ contains all the information on when and how $\tau$ has happened.

To give a little bit more insight consider this slight modification on m7e's example: let $(X_n)_{n \in \{1,\dots,4\}}$ be a an iid sequence such that $\mathbb P(X_1=1)=\mathbb P(X_1=-1)= \frac 12$ and define $S_n= \sum_{k=1}^n X_k$. Intuitively, we can think about this as a game where each round a fair coin is tossed and I either win $1\$$ (if $X_n=1$) or lose $1\$$ (if $X_n=-1$). Also I define a stopping time $\tau$ as $$ \tau:= \inf \{n \geq 2\mid X_{n-1}=1 \ \text{and} \ X_n=1\} \wedge 5. $$ Following the stopping time $\tau$ means that I stop playing after I win two times in a row or at the end of the fifth round at latest.

For the following I will be using the following (simple but sloppy) notation, where $W$ is short for win and $L$ is short for lose: \begin{align} A_{WL}&= \{\omega \in \Omega \mid X_1(\omega)=1 \ \text{and} \ X_2(\omega)=-1 \}, \quad \text{or}\\ A_{LLW}&=\{\omega \in \Omega \mid X_1(\omega)=X_2(\omega)=-1 \ \text{and} \ X_3(\omega)=1 \}, \quad \text{etc} \end{align}.

Then the $\sigma$-algebra generated by $\tau$ is generated by the following sets: $$ \sigma(\tau)=\sigma \Bigl\{A_{WW}, \ A_{LWW},\ A_{WLWW}\cup A_{LLWW} \Bigr\}. $$ Here, it is important to note that neither of the sets $A_{WLWW}$ and $A_{LLWW}$ are contained in $\sigma(\tau)$ on there own. Put differently, given the information contained in $\sigma(\tau)$ I can distinguish between the events $\{\tau=2\}$ (which corresponds to $A_{WW}$), $\{\tau=3\}$ (which corresponds to $A_{LWW}$) and $\{\tau=4 \}$ (which corresponds to $A_{WLWW}\cup A_{LLWW}$) but I cannot distinguish between $A_{WLWW}$ and $A_{LLWW}.$

This means that $\sigma(\tau)$ tells me when I stopped playing the game, but not the history of the game which has led to the event of me stopping. In particular, if I know that I stopped after four rounds, I do not know what happened in the first two rounds.

To sum up $\sigma(\tau)$ contains all the information on when $\tau$ has happened but not on how $\tau$ has happened.

On the other hand, $\mathcal F_\tau$ contains the following sets: $$ \mathcal F_\tau=\sigma \Bigl\{A_{WW}, \ A_{LWW},\ A_{WLWW}, A_{LLWW} \Bigr\}. $$ which allows a differentiation between $A_{WLWW}$ and $A_{LLWW}$. So given the information contained in $\mathcal F_\tau$ and I know when I stopped playing the game and in addition the history up to the stopping of the game.