= Solution
Apply the <realized absolute covariation> theorem to each dyadic <partition of an interval>. More explicitly, use the continuous increasing clock $C=[M]+[N]$ and the <Radon-Nikodym theorem> to write
$$
a=\frac{d[M]}{dC},\qquad b=\frac{d[N]}{dC},\qquad c=\frac{d[M,N]}{dC}.
$$
For each time, let $(U,V)$ have the centered <bivariate normal distribution> with covariance matrix $\left(\begin{smallmatrix}a&c\\c&b\end{smallmatrix}\right)$, and define
$$
\widetilde V_t=\int_0^t\mathbb E|UV|\,dC.
$$
This process is continuous and increasing. To prove convergence, localize $M,N$, represent the pair as <stochastic integrals> against a two-dimensional <Brownian motion>, and approximate the integrands in $L^2(dC)$ by bounded step <previsible processes>. For step integrands, the result is the <weak law of large numbers> applied on each block to independent <Gaussian random variables>. The <Burkholder-Davis-Gundy inequality> and the <Cauchy-Schwarz inequality> make the error uniform on each compact interval in probability. Consequently
$$
\widetilde V^n\longrightarrow\widetilde V
$$
in the sense of <uniform convergence on compacts in probability>.
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