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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.

Fabrice Baudoin studies stochastic calculus and differential geometry, using diffusion processes and heat kernels to obtain functional inequalities. His Aarhus University research profile describes this exchange: geometric information can bound a diffusion, while probabilistic methods can reveal properties of the underlying space.^[ref]

His lecture Itō's representation theorem by Fabrice Baudoin illustrates the role of a precise information structure. For the completed natural filtration of Brownian motion, a square-integrable terminal random variable can be written as its expected value plus a stochastic integral. This yields a continuous version of each square-integrable martingale in that filtration. His exposition develops the Brownian martingale representation theorem through its filtration and square-integrability assumptions.^[ref]