Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 2

A process is previsible when it is measurable with respect to the predictable sigma-algebra. The deterministic process $H_t=1_{[1,\infty )}(t)$ is predictable because deterministic Borel processes are predictable, but it is not left-continuous at $t=1$ on any sample path.

Since $|Z_t{Y}_t|\le C{Z}_t$ and $Z$ is uniformly integrable, the continuous local martingale $ZY$ is locally in Doob's class and hence is a true martingale. For $A\in {\mathcal{F}}_s$ and $s\le t$, Bayes formula for conditional expectation givesThe bounded process $Y$ is integrable under $\tilde{\mathbb{P}}$, so this is exactly the martingale property.

The Cameron-Martin theorem on Wiener space says that the translated measure ${\mathbb{P}}_h(A)=\mathbb{P}(X+h\in A)$ is equivalent to Wiener measure precisely when $h$ is absolutely continuous, $h(0)=0$, and $\dot{h}\in L^2({\mathbb{R}}_{+})$. In that case

For such $h$, the exponential is a uniformly integrable stochastic exponential. Under the measure defined by this density, the Girsanov theorem makes $X_t-{\int }_0^t{\dot{h}}_{s}ds=X_t-h(t)$ a Brownian motion. This identifies the translated law and proves equivalence; replacing $h$ by $-h$ gives the inverse density.

If $h$ fails the Cameron-Martin condition on some finite interval, the finite-horizon theorem gives singularity there. If it belongs locally but ${\int }_0^{\infty }{\dot{h}}_s^{2}ds=\infty$, the log likelihood is a Brownian motion run at that diverging energy clock minus half the clock. It tends to $-\infty$ under one measure and to $+\infty$ under the translate, producing disjoint full-measure events. Thus the measures are singular.

The Cameron-Martin theorem for a linear drift changes the density of Brownian paths through time $t$ byAt the driftless hitting time ${\tau }_{a,0}=t$, the endpoint is $B_t=a$. Multiplying its given density by the likelihood $e^{ab-b^{2}t/2}$ therefore yields