Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 26K (Stochastic Financial Models)

A martingale relative to a filtration $({\mathcal{F}}_n)$ is an adapted sequence of integrable random variables with $E[X_{n+1}\mid {\mathcal{F}}_n]=X_n$ almost surely for every $n$. Adapted means $X_n$ is ${\mathcal{F}}_n$-measurable. Iterating gives $E[X_m\mid {\mathcal{F}}_n]=X_n$ for all $m\ge n$.

Conditional expectation makes $Y_n$ measurable with respect to ${\mathcal{F}}_n$, and conditional Jensen gives $E|Y_n|\le E|Y|<\infty$. The tower property of conditional expectation for nested sigma-algebras givesThus this is a martingale. Its finite limit exists almost surely for every integrable $Y$ under the stated increasing filtration: the martingale convergence theorem applies to the uniform bound on $E|Y_n|$. In fact the family is uniformly integrable. One way to see this is to approximate $Y$ in $L^1$ by a bounded variable, whose conditional expectations remain bounded, while the conditional expectations of the error have uniformly small $L^1$ norm. Consequently convergence is also in $L^1$, with limit $E[Y\mid \sigma ({\bigcup }_n{\mathcal{F}}_n)]$. Mere martingale status without a suitable bound would not imply convergence.

Use the natural urn filtration. At stage $n$, put $s=X_n+Y_n=n+2$ and $d=X_n-Y_n$. The difference becomes $d-1$ with probability $X_n/s$ and $d+1$ with probability $Y_n/s$. Hence its conditional next mean is $d+(Y_n-X_n)/s=d(1-1/s)$. Since the next total is $s+1$,Adaptedness is immediate, and every $M_n$ is bounded by a deterministic finite number, so integrable. This proves the martingale property.

Nevertheless it does not converge to a finite limit on any sample path. The increments are $d+s$ or $d-s$, because $M_{n+1}=s(d\pm 1)$ and $M_n=(s-1)d$. Both colours remain present, so $|d|\le s-2$ and $|M_{n+1}-M_n|\ge 2$. Convergence of a real sequence would force its increments to tend to zero. There is therefore no almost-sure finite convergence, illustrating why the boundedness condition in the preceding part matters.