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

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