= Natural Brownian filtration
{title2=$\mathcal F_t^W=\sigma(W_s:0\leq s\leq t)$}
The natural Brownian filtration is the <natural filtration> generated by a <Brownian motion>. Its usual augmentation adds null events and makes it right-continuous; the <Brownian martingale representation theorem> holds for this augmented natural filtration. A larger <Brownian filtration> can preserve independence of future increments while containing additional randomness, so it need not have martingale representation with respect to the specified <Brownian motion>.
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