Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 39 / 4 / i / Solution

Treat the two-point prior for $\alpha$ as independent of the driving Brownian motion. On a partition $0=t_0<\cdots <t_n=t$, conditionally on $\alpha =\pm a$, the increments of $X$ are independent normal distributions with means $\pm a\Delta t_j$ and variances $\Delta t_j$. Their likelihood ratio isThus every partition containing $t$ gives the same Bayesian posterior, depending only on the endpoint. Refine partitions through a dense set of observation times. Continuity makes their increasing observation fields generate the full path field ${\mathcal{X}}_t$, and convergence of bounded conditional expectations preserves that Bayesian posterior. Equal prior probabilities and the Bayes' theorem consequently giveIts conditional mean drift isThis is the binary Brownian drift filter.

Define the observed innovation processIt is continuous and adapted to ${\mathcal{X}}_t$. For $s<t$, independence of future Brownian motion increments from the past and the latent drift givesFor every $u\ge s$, the tower property of conditional expectation givesSince $|h|\le a$, conditional integration is justified. Subtracting the integrated identities shows $E[{\hat{W}}_t-{\hat{W}}_s\mid {\mathcal{X}}_s]=0$. Thus $\hat{W}$ is a square-integrable martingale. Subtracting a continuous finite variation process does not change quadratic variation, so $[\hat{W}{]}_t=[X{]}_t=[W{]}_t=t$. The Lévy characterization of Brownian motion now proves that $\hat{W}$ is a Brownian motion in the observation filtration. Therefore