Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2008 / A / 3

Fix a finite horizon $T$ and use the completed natural Brownian filtration. Martingale representation theorem states that every $F\in L^2({\mathcal{F}}_T)$ can be writtenwhere $H$ is predictable and is unique up to equality for $dsd\mathbb{P}$ almost every $(s,\omega )$.

The Martingale representation theorem states that every square-integrable martingale $M$ in this filtration has a continuous version of the formwith the same integrability and uniqueness condition. The completed natural filtration has only null-set randomness at time zero, so $M_0$ is almost surely constant. The first statement gives the second by representing $M_T$ and taking conditional expectations; the second gives the first by taking $M_t=\mathbb{E}(F\mid {\mathcal{F}}_t)$.

There is also the local version: every local martingale in the Brownian filtration has a continuous version representable by a predictable, locally square-integrable integrand. The filtration restriction matters; an enlarged filtration with independent additional martingales need not have representation using $B$ alone.

For $f(z)=\sin z$, $f^{\prime }(z)=\cos z$ and $f^{\prime \prime }(z)=-\sin z$. Itô's formula givesSince $B_0=0$, this is exactlyThe integrand retains $\cos B_t$; replacing it by a fixed sign of $\sqrt{1-X_t^2}$ would lose information about the phase of $B_t$.

Condition on the Brownian path up to $t\le T$. The independent normal increment $B_T-B_t$ has $\mathbb{E}\cos (B_T-B_t)=e^{-(T-t)/2}$ and zero expected sine. HenceApply Itô's formula to this expression. Its time derivative and second spatial derivative cancel, leavingTherefore the required representation isThe bounded integrand is square integrable, and uniqueness follows from the Itô isometry. The drift-containing decomposition in part (b) alone would not be the required representation as a constant plus one stochastic integral.