Use this tag for a general type of quadratic variation of two stochastic processes.
Suppose that $X_t$ and $Y_t$ are a real-valued stochastic processes defined on a probability space $(\Omega, \mathcal F, \mathbb P)$ and with time index $t$ ranging over non-negative real numbers. The cross-variation of $X$ and $Y$ is $$\langle X, Y \rangle_t = \lim_{\Vert P \Vert \to 0} \sum_{k=1}^n \left(X_{t_k} - X_{t_{k-1}} \right) \left( Y_{t_k} - Y_{t_{k-1}} \right)$$ where $P$ ranges over partitions of the interval $[0, t],$ and the norm of the partition $P$ is the mesh. The limit, if it exists, is defined using convergence in probability.
Cross-variation may be written in terms of quadratic variation by the polarization identity $$\langle X, Y \rangle_t = \tfrac12 (\langle X + Y \rangle_t - \langle X \rangle_t - \langle Y \rangle_t).$$