Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / A / 3 / b

The printed update needs a one-step index correction to match its filtration: the draw for the transition from $n$ to $n+1$ must be the fresh variable $U_{n+1}$. Also the trivial initial sigma-algebra is $\{\varnothing ,\Omega \}$, rather than the printed $\{\varnothing ,\mathcal{F}\}$. With literal $U_n$, the next increment is already known at time $n$, so $\mathbb{E}[X_{n+1}\mid {\mathcal{F}}_n]=X_{n+1}$; the nontrivial update differs from $X_n$. Thus the stated martingale claim is false with that indexing. The following uses the standard Pólya urn described in words.

Put $m=B_0+R_0$, so the total count is $m+n$. Conditional on ${\mathcal{F}}_n$, the next blue indicator $I_{n+1}$ has expectation $X_n$. HenceAdaptedness and integrability follow from the recursive construction and $0\le X_n\le 1$. Thus $X_n$ is a bounded martingale. The martingale convergence theorem gives an almost-sure limit $X_{\infty }\in [0,1]$. Boundedness and the dominated convergence theorem then give $\mathbb{E}|X_n-X_{\infty }{|}^2\to 0$, establishing convergence in $L^2$.

The blue count never decreases, so $X_n\ge B_0/(m+n)$. For all sufficiently large $n$, this is at least $B_0/[2(n+1)]$. Whenever the fresh draw satisfies $U_{n+1}\le B_0/[2(n+1)]$, a blue ball is added. Part (a), with $\lambda =B_0/2$, says that such events occur infinitely often almost surely. Therefore $B_n\to \infty$. Consequently

Since $X_n>0$, the pointwise inequalityholds: on that event both sides equal one, and elsewhere the left side is zero. Take conditional expectation and pull out the positive ${\mathcal{F}}_n$-measurable denominator to obtain the requested bound. The supplied variance estimate improves it toalmost surely, by part (ii). These conditional probabilities are bounded by one. The dominated convergence theorem and tower property of conditional expectation giveThis proves Positivity of the limiting Pólya urn proportion without needing its explicit beta distribution.