Martingale representation theorem

A martingale representation theorem identifies driving martingales from which every martingale in a specified filtration can be reconstructed by stochastic integrals. The filtration and the class of integrable martingales are part of the assertion.

For the completed natural Brownian filtration, the Brownian martingale representation theorem gives $M_t=M_0+{\int }_0^t{H}_{s}d{W}_s$ for every square-integrable martingale, with a predictable integrand $H$ unique up to time-probability null sets. For example, $W_t^2-t={\int }_0^{t}2W_{s}d{W}_s$. In a filtration generated by independent Rademacher random variables, predictable representation in a Rademacher filtration gives the discrete analogue. Adding an independent source of randomness to a Brownian filtration may require another driving martingale; it does not preserve representation by the original Brownian motion alone.

The conditional expectation of $\sin B_T$ at time $t$ is $e^{-(T-t)/2}\sin B_t$, by independent Brownian increment. Itô's formula cancels its time derivative against its second spatial derivative and leaves the displayed stochastic integral. The integrand is bounded, so Itô isometry verifies its square integrability.

Let $({\mathcal{F}}_t)$ be the usual augmentation of the natural filtration of a standard Brownian motion $W$. On a finite horizon $T$, every $Z\in L^2({\mathcal{F}}_T)$ has the representationfor a predictable process $H$. Consequently every square-integrable martingale on $[0,T]$ has $M_t=M_0+{\int }_0^t{H}_{s}d{W}_s$. The Itô isometry makes $H$ unique up to $dtd\mathbb{P}$-almost everywhere equality. A proof first represents deterministic Gaussian exponentials, whose linear span is dense in L2 space, then uses the Itô isometry to close the class of represented terminal values.^[ref]

The natural-filtration qualification matters. In the filtration generated by independent Brownian motions $W,V$, both are martingales, but $V$ cannot be an Itô integral against $W$: its quadratic covariation with $W$ is zero, which would force the proposed integrand to vanish. A general Brownian filtration may therefore be too large.

In the completed natural Brownian filtration, a bounded continuously differentiable $\varphi$ with bounded derivative satisfiesThe integrand is unique up to $d\mathbb{P}dt$-almost everywhere equality. Gaussian integration by parts first gives the duality between the payoff and every square-integrable Itô integral. The Brownian martingale representation theorem then identifies the integrand as the projection of ${\varphi }^{\prime }(W_T)1_{\{t\le T\}}$ onto the predictable processes in the product $L^2$ space. This explains the conditional expectation in the formula, rather than a nonadapted terminal derivative.