Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 202 / 4 / d / Solution

The Lévy characterization of Brownian motion states that an adapted continuous real process $M$ with $M_0=0$ is standard Brownian motion relative to $({\mathcal{F}}_t)$ if and only if it is a continuous local martingale and $[M{]}_t=t$. The multidimensional version replaces the bracket identity by $[M^i,M^j{]}_t={\delta }_{ij}t$.

For the converse direction, a standard Brownian motion is a continuous martingale, and its independent centered increments give quadratic variation $t$. Distinct independent coordinates have zero quadratic covariation.

For the substantive direction, fix a vector $\theta \in {\mathbb{R}}^d$. The Itô formula applied toshows that its drift is zero: the second-order term is $-\frac{1}{2}|\theta {|}^2{E}_{t}dt$ and cancels the time derivative. Thus $E(\theta )$ is a complex local martingale. On $[0,T]$ its modulus is the deterministic value $e^{|\theta {|}^{2}t/2}\le e^{|\theta {|}^{2}T/2}$; localization and the dominated convergence theorem make it a true martingale. HenceThis conditional characteristic function is the deterministic characteristic function of $N(0,(t-s)I)$. For $A\in {\mathcal{F}}_s$, multiplying by $1_A$ and taking expectations, then using uniqueness of characteristic functions, proves that $M_t-M_s$ has this multivariate normal distribution and is independent of ${\mathcal{F}}_s$. Iterated conditioning gives independent increments with Gaussian distributions. Continuity, the zero initial value and these increment laws are the definition of standard Brownian motion. The case $d=1$ proves the requested statement; the vector form will also be used below.