= Realized absolute covariation
{title2=$\widetilde V_t$}
For continuous local martingales $M,N$ and a sequence in which each term is a <partition of an interval> whose mesh tends to zero, the sums
$$
\widetilde V_t^n=\sum_k|\Delta_kM|\,|\Delta_kN|
$$
converge in the sense of <uniform convergence on compacts in probability> to a continuous increasing process. To identify the limit, put $C=[M]+[N]$ and choose <Radon-Nikodym derivatives>
$$
a=\frac{d[M]}{dC},\qquad b=\frac{d[N]}{dC},\qquad c=\frac{d[M,N]}{dC}.
$$
If $(U,V)$ is a centered <bivariate normal distribution> with covariance matrix $\left(\begin{smallmatrix}a&c\\c&b\end{smallmatrix}\right)$, then
$$
\widetilde V_t=\int_0^t\mathbb E|UV|\,dC.
$$
Localizing, representing the pair as <stochastic integrals> against a two-dimensional <Brownian motion>, and approximating the integrands by bounded predictable step processes proves the convergence. The step-process case follows from the <weak law of large numbers> for independent Gaussian increments; the <Burkholder-Davis-Gundy inequality> controls the approximation error. Since $|\mathbb E[UV]|\leq\mathbb E|UV|\leq\sqrt{\mathbb EU^2\mathbb EV^2}$,
$$
V_t([M,N])\leq\widetilde V_t\leq[M]_t^{1/2}[N]_t^{1/2}.
$$
Back to article page
