Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 202 / 5

A simple predictable process has the formwhere each bounded ${\xi }_i$ is ${\mathcal{F}}_{t_i}$-measurable. DefineIndependent centered Brownian increments show directly by conditioning that this is a martingale. The same conditional expansion, using $\mathbb{E}[(B_v-B_u{)}^2\mid {\mathcal{F}}_u]=v-u$, shows that

Since $B_t={\int }_0^{t}1d{B}_s$, the Itô isometry and polarization give

With $A_t={\int }_0^t\mu (s)ds$, the product rule givesso $X_t{e}^{-A_t}$ is a local martingale under $\mathbb{P}$.

Set $\theta (t)=\mu (t)/\sigma (t)$. This is bounded and compactly supported, so Novikov condition holds anddefines a probability measure $d\mathbb{Q}=Z_{\infty }d\mathbb{P}$. By the Girsanov theorem, $W_t=B_t+{\int }_0^t\theta (s)ds$ is Brownian under $\mathbb{Q}$, andThus $X$ is a local martingale under $\mathbb{Q}$.