Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 3

Write the $d$-dimensional continuous semimartingale as $X=X_0+M+A$, where $M$ is a continuous local martingale and $A$ is a continuous finite-variation process. For $f\in C^2({\mathbb{R}}^d)$, Itô formula states

The $d$-dimensional Lévy characterization of Brownian motion says that a continuous local martingale $X$ with $X_0=0$ is a standard $d$-dimensional Brownian motion exactly whenfor all $i,j$ and $t$.

One direction follows directly from independent Gaussian increments. Conversely, fix $\theta \in {\mathbb{R}}^d$. Applying Itô formula and the bracket assumption shows thatis a complex local martingale. After stopping $X$ on leaving large balls it is bounded, so optional sampling and then dominated convergence give, for $s<t$,This is the characteristic function of $N(0,(t-s)I_d)$ and is deterministic. Thus each increment is Gaussian with the required covariance and independent of the past. Together with continuity, these are precisely the defining properties of standard $d$-dimensional Brownian motion.

Put ${\theta }_t=a{t}^{a-1}$. When $a>1/2$,The integrand is deterministic, so the Novikov condition holds. Define an equivalent probability measure $Q$ byThe Cameron-Martin-Girsanov theorem makesa $Q$-Brownian motion on $[0,T]$.

The given limit inferior says that almost surely there is a random $\delta >0$ such that $t^{-a}{W}_t>-1/2$ whenever $0<t\le \delta$. Thereforeon that interval. The first entrance timeis a stopping time by continuity and satisfies $\tau >0$ almost surely. Before $\tau$ one has $S_t>t^a/2$, and continuity gives $S_{\tau }\ge {\tau }^a/2>0$. Hence $S_t>0$ for every $0<t\le \tau$.

Suppose an equivalent measure $Q$ made $S$ a local martingale. Stop at the positive time furnished by part (i), chosen so that $S_{t\wedge \tau }\ge 0$. A nonnegative local martingale is a supermartingale, and this one starts from $S_0=0$. Hence $S_{t\wedge \tau }=0$ $Q$-almost surely for every deterministic $t$.

But $Q$ and $P$ have the same null events, while $\tau >0$ almost surely. Some positive rational $q$ therefore satisfies $Q(q<\tau )>0$, and on that event part (i) gives $S_q>0$, a contradiction. No such equivalent measure exists.