= Brownian martingale representation theorem
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= Itô martingale representation theorem
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{synonym}
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 representation
$$
Z=\mathbb EZ+\int_0^TH_s\,dW_s,\qquad \mathbb E\int_0^TH_s^2ds<\infty,
$$
for a <predictable process> $H$. Consequently every <square-integrable martingale> on $[0,T]$ has $M_t=M_0+\int_0^tH_s\,dW_s$. The <Itô isometry> makes $H$ unique up to $dt\,d\mathbb P$-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.<Itō's representation theorem by Fabrice Baudoin>[2012 lecture, theorem and proof by density of stochastic exponentials]{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.
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