Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 5

The relation $P\ll \tilde{P}$ is an instance of absolute continuity of measures: it means that every $\tilde{P}$-null event is also $P$-null,By the Radon-Nikodym theorem, this is equivalent to the existence of a nonnegative Radon-Nikodym derivative $dP/d\tilde{P}$ whose $\tilde{P}$-expectation is one.

For a continuous local martingale $M$ with $M_0=0$, its stochastic exponential isIt is the unique solution of the stochastic differential equation $d{Z}_t=Z_{t}d{M}_t$, $Z_0=1$, and is a nonnegative local martingale.

One continuous form of the Girsanov theorem is as follows. Let $M$ be a continuous local martingale under $P$, and suppose $Z=\mathcal{E}(M)$ is a true martingale on $[0,T]$. Define $Q$ by $dQ=Z_{T}dP$. Then every continuous $P$-local martingale $N$ becomes the continuous $Q$-local martingaleIn particular, if $M_t={\int }_0^t{\theta }_{s}d{W}_s$, then ${\tilde{W}}_t=W_t-{\int }_0^t{\theta }_{s}ds$ is a $Q$-Brownian motion.

Work first under Wiener measure $P$ with coordinate Brownian motion $X$. Boundedness of $b$ implies the Novikov condition, sohas expectation one. Define $Q$ by $dQ=Z_{T}dP$. The Girsanov theorem makesa $Q$-Brownian motion, and hence $(X,W,Q)$ is a weak solution of a stochastic differential equation.

For uniqueness in law, start with any weak solution under $Q$ and apply the inverse change of measure with density $\mathcal{E}(-\int b(X_s)d{W}_s{)}_T$. Boundedness again gives the Novikov condition, and under the resulting measure $P$ the process $X$ is Brownian. Reversing the density expresses the law of $X$ under $Q$ as the same functional $Z_T$ of a Wiener path. It is therefore independent of the chosen weak solution. This proves the Weak existence and uniqueness in law for an additive-noise SDE with bounded drift.

The assertion is false. For a common sequence, each term of which is a refining deterministic partition of an interval, standard Brownian paths have quadratic variation $t$ almost surely, whereas the paths $t\mapsto B_{2t}$ have quadratic variation $2t$ almost surely. These two path properties define disjoint measurable subsets of $C([0,1])$, so the two laws are mutually singular measures. In particular, the law of $B_{2\cdot }$ is not absolutely continuous with respect to Wiener measure. This is the Pathwise quadratic variation distinguishes Brownian speeds argument.

The assertion is true. Since $f\in C^1[0,1]$, it lies in the Cameron-Martin space of Wiener measure. The Cameron-Martin theorem says that the translated law is equivalent, and in particular absolutely continuous, with respect to Wiener measure. Its Radon-Nikodym derivative is

The assertion is false for a general continuous $f$. For example, take $f(t)=t^{1/3}$. The Brownian Hölder regularity gives $B_t/t^{1/3}\to 0$ almost surely as $t\downarrow 0$, whilealmost surely. The original and translated path laws therefore concentrate on disjoint measurable events and are mutually singular measures.

More generally, the Cameron-Martin theorem gives the exact criterion: translation by $f$ is absolutely continuous precisely when $f$ is an absolutely continuous function, $f(0)=0$, and $f^{\prime }\in L^2[0,1]$.