Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / A / 1

A filtration is an increasing sequence of sub-sigma-algebras ${\mathcal{F}}_n\subseteq \mathcal{F}$: if $m\le n$, then ${\mathcal{F}}_m\subseteq {\mathcal{F}}_n$. It represents information available through each time.

A stopping time $T:\Omega \to {\mathbb{N}}_0\cup \{\infty \}$ satisfiesThus whether stopping has already occurred is determined by the current information.

The whole space belongs to ${\mathcal{F}}_T$, since $\Omega \cap \{T\le n\}=\{T\le n\}\in {\mathcal{F}}_n$. If $A\in {\mathcal{F}}_T$, thenIf $A_j\in {\mathcal{F}}_T$, then $({\bigcup }_j{A}_j)\cap \{T\le n\}={\bigcup }_j(A_j\cap \{T\le n\})$ is also in ${\mathcal{F}}_n$. These are precisely the sigma-algebra closure properties, so ${\mathcal{F}}_T$ is a sigma-algebra.

For every integer $m\ge 0$,Hence $\{T\le m\}\in {\mathcal{F}}_T$. Such threshold sets generate the Borel sigma-algebra on ${\mathbb{N}}_0\cup \{\infty \}$, proving $T$ is ${\mathcal{F}}_T$-measurable. This is the stopping-time sigma-algebra.

For $A\in {\mathcal{F}}_S$, writeThe first two factors together belong to ${\mathcal{F}}_k$, since $A\cap \{S=k\}$ is the difference of its intersections with $\{S\le k\}$ and $\{S\le k-1\}$. Also $\{k\le T\le n\}\in {\mathcal{F}}_n$ by the stopping time property. The filtration inclusion puts every term, and therefore the union, in ${\mathcal{F}}_n$. By definition,

If $S\le T$, then $\{S\le T\}=\Omega$. The previous result says that every $A\in {\mathcal{F}}_S$ equals $A\cap \{S\le T\}\in {\mathcal{F}}_T$. ThusInformation known at the earlier stopping time remains known at the later stopping time.

Both stopped variables are integrable, since $|X_S|,|X_T|\le {\sum }_{j=0}^N|X_j|$. Also $X_S$ is ${\mathcal{F}}_S$-measurable: intersecting $\{X_S\in B\}$ with $\{S\le n\}$ gives the finite union ${\bigcup }_{j\le \min (n,N)}\{S=j,X_j\in B\}$ in ${\mathcal{F}}_n$ for each Borel $B$.

The increment identity isFor $A\in {\mathcal{F}}_S$, the event $A\cap \{S\le k\}\cap \{T>k\}$ belongs to ${\mathcal{F}}_k$. The martingale increment has conditional mean zero given that sigma-algebra. Multiplying by its indicator and using the tower property gives zero expectation for each summand. ThereforeTogether with measurability and integrability, this is the defining property of conditional expectation, proving the bounded optional sampling theorem