= Quadratic-variation measure
{title2=$\nu_M$}
For a <L2-bounded continuous martingale> $M$, define a finite <measure> on the <predictable sigma-algebra> by $\nu_M(A)=\mathbb E\int_0^\infty\mathbf1_A\,d[M]$. Its total mass is $\mathbb E[M]_\infty=\mathbb E M_\infty^2-\mathbb E M_0^2$. The <Lebesgue space> $L^2(\nu_M)$ identifies <predictable processes> that agree $\nu_M$-almost everywhere. Integration against $M$ is an isometry from this space into <L2-bounded continuous martingales> starting at zero by the <Itô isometry>. This gives the correct integrand space even when the <quadratic variation> is random or is not absolutely continuous in time.
Back to article page
