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I'm having problems understanding the concepts of measurability and stopping times. Here's the problem I'm having issues with $\to$

Let $\{X_i\}_{i\geq1}$ be a sequence of independent and identically distributed random variables with $\mathbb{P}(X_i=1)=p, \mathbb{P}(X_i=-1)=1-p$, where $0<p<1$ and $p \neq 0.5$. Let $$S_n=\sum_{i=1}^{n}X_i $$

and

$$M_n=\left( \frac{1-p}{p} \right)^{S_n}.$$

Moreover, let $\{\mathcal{F}_n \}_{n \geq 1}$ be the natural filtration defined by $\mathcal{F}_n=\sigma(X_1, \dots,X_n)$ for every $n \geq 1$.

(a) Show that $\{M_n\}_{n \geq 1}$ is a martingale with respect to $\{\mathcal{F}_n\}_{n \geq 1}$.

(b) Let $A$ be an $\mathcal{F}_m$-measurable event for some $m$, $A^C$ be the complement of $A$ and let the constants $a,b \geq m$. Show that $\mathbb{E}[M_{\tau}]=\mathbb{E}[M_m]$ where $\tau=a1_A+b1_{A^C}$.

So, I guess I'm ok with the (a) part but I have some problems with understanding the (b) part.

  1. So, the fact that $A$ is $\mathcal{F}_m$-measurable event means that $A \in \mathcal{F}_m$. But what does that actually mean? Does it mean that at the time $t=m$ we will definitely know if the event $A$ has happened or not, or does it just mean that at time $t=m$ we will just "observe" event $A$ for the first time, i.e. include it in our $\sigma$-algebra and assign some probability to it? This (latter) point of view makes more sense to me, but it leads to some confusion when the stopping times come into the picture...

  2. Now, the introduction of stopping times has just made all of this more confusing to me. Like, if $\mathcal{F}_m$ is the first $\sigma$-algebra in our filtration containing event $A$, how do we calculate $\tau=a1_A+b1_{A^C}$ for $\omega \in \Omega$ at time $t=1$? How can we tell at time $t=1$ if any $\omega$ is in $A$ or $A^C$ if $A$ is not measurable at that moment and won't be measurable (am I mixing up the notions of measurable and observable) before time $t=m$? Is it that until time $t=m$ the values of $1_A$ and $1_{A^C}$ will be equal to $0$ for all $\omega$?

I would appreciate any help with this obviously basic but crucial concepts. Thanks!

Milan
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  • My answer here maybe helpful: http://math.stackexchange.com/questions/331410/why-is-stopping-time-defined-as-a-random-variable/331474#331474 –  Dec 11 '15 at 21:05

1 Answers1

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  1. $A\in{\cal F}_m$ means that we can tell whether or not $\omega\in A$ by looking only at $X_0(\omega)$,$X_1(\omega)$, $\dots$ ,$X_m(\omega)$; the subsequent values $X_{m+1}(\omega), \dots$ are not needed. More formally, there is some measurable set $B$ of $\mathbb{R}^{m+1}$ such that $$A=\{\omega: (X_0(\omega),X_1(\omega), \dots ,X_m(\omega))\in B\}.$$

  2. For part 2, $M_\tau= M_a 1_A+M_b 1_{A^c},$ so that when you take conditional expectation with respect to ${\cal F}_m$ you get $$\mathbb{E}(M_\tau\mid{\cal F}_m)=\mathbb{E}(M_a\mid{\cal F}_m)\,1_A +\mathbb{E}(M_b\mid{\cal F}_m)\,1_{A^c}=M_m.$$ I leave you to fill in the details and finish the exercise.