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

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.