Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 30 / 3

A scalar process is a Brownian motion in $({\mathcal{F}}_t)$ when it is adapted, has continuous sample paths, starts at zero almost surely, and, for every $0\le s<t$,Independence from the whole past sigma-field is the essential filtration requirement. It implies independent increments over disjoint intervals; the variance $t-s$ fixes the standard Brownian time scale.

Fix $s<t$ and an event $A\in {\mathcal{F}}_s$. Multiplying the conditional characteristic function identity by $1_A$ and taking expectation givesThe left side is the Fourier transform of the finite measureBy the uniqueness theorem for characteristic functions, this measure equals $\mathbb{P}(A)$ times the $N(0,t-s)$ distribution. Thus, for every Borel set $D$,Taking $A$ to be the whole sample space identifies the increment's law, and the full identity proves independence from ${\mathcal{F}}_s$. Together with the assumed adaptation, continuity, and initial value, these are exactly the defining Brownian properties. This proves the conditional characteristic-function criterion for Brownian increments.

The scalar Lévy characterization of Brownian motion states that an adapted process $X$ starting at zero is a Brownian motion if and only if it is a continuous local martingale with

For necessity, the centered independent increments make a Brownian motion a martingale. Their conditional second moments show that $X_t^2-t$ is also a martingale. The defining uniqueness of the quadratic variation compensator gives $⟨X{⟩}_t=t$.

For sufficiency, fix $\theta \in \mathbb{R}$ and apply the Itô formula toThe time drift cancels the second-order Itô term, leaving $d{Z}_t=i\theta Z_{t}d{X}_t$. Its real and imaginary parts are local martingales. On any deterministic interval $[0,T]$, $|Z_t|=e^{{\theta }^{2}t/2}\le e^{{\theta }^{2}T/2}$, so the bounded local martingale criterion makes them true martingales. ThereforePart (b) proves the Brownian property. This also supplies a proof of Lévy's characterization of Brownian motion through conditional Fourier transforms.

Because $K$ is continuous, adapted, and strictly positive, $K^{-1/2}$ is predictable and locally integrable against $M$. DefineThe quadratic variation of a stochastic integral isThus $W$ starts at zero and is a continuous local martingale with bracket $t$. The Lévy characterization of Brownian motion makes it a Brownian motion in the original filtration.

The process $H$ is continuous and adapted, and ${\int }_0^t{H}_s^{2}ds={\int }_0^t{K}_{s}ds<\infty$ almost surely. The associativity of stochastic integration givesThis is the stochastic-integral representation from absolutely continuous quadratic variation.