Quadratic covariation ($[M,N{]}_t$)

The quadratic covariation of two continuous semimartingales is the limit in probabilityThe polarization identity gives $[M,N]=\frac{1}{4}([M+N]-[M-N])$.

For an analytic function $f=u+iv$ and planar Brownian motion $Z=X+iY$ with independent standard coordinates, the Cauchy-Riemann equations make $u,v$ harmonic. The Itô formula gives $dU=u_{x}dX+u_{y}dY$ and $dV=-u_{y}dX+u_{x}dY$. Localizing the derivatives proves both coordinates are local martingales. Squaring their diffusion coefficients and multiplying the two rows gives the displayed quadratic variations and zero quadratic covariation.

For two zero-starting continuous martingales square-integrable at time $t$, the Itô isometry gives $\mathbb{E}(M_t\pm N_t{)}^2=\mathbb{E}[M\pm N{]}_t$. The polarization identity yields the displayed identity. Integrability of the cross-bracket follows from the Kunita-Watanabe inequality and the Cauchy-Schwarz inequality.

For zero-starting L2-bounded continuous martingales, this condition is equivalent to $\mathbb{E}[M_t{N}_t]=0$ at equal times, by conditioning the later factor. The covariance identity for continuous square-integrable martingales also makes it equivalent to $\mathbb{E}[M,N{]}_t=0$. It is weaker than the vanishing bracket condition for orthogonal continuous local martingales: $B_{t\wedge 1}$ and $B_{t\wedge 1}^2-(t\wedge 1)$ have zero equal-time covariance but nonzero bracket $2{\int }_0^{t\wedge 1}{B}_{s}ds$.

Two continuous local martingales are orthogonal when their quadratic covariation is zero. With zero initial values, the martingale product identity says that their product is a local martingale. For two Brownian motions in a common filtration, orthogonality and the Lévy characterization of multidimensional Brownian motion make the pair a two-dimensional Brownian motion and imply independence. Merely having the two marginal Brownian motion laws on the same space does not imply orthogonality.

Let $M_1,\dots ,M_N$ be continuous local martingales starting at zero, with $⟨M_i,M_j⟩=0$ for $i\ne j$ and $⟨M_j{⟩}_{\infty }=\infty$ almost surely. The individual Dambis-Dubins-Schwarz theorem time changes produce independent standard Brownian motions $B_j$ with $M_j(t)=B_j(⟨M_j{⟩}_t)$. Independence is between the Brownian motions; each may still depend on its own clock.

For orthogonal continuous local martingales $B,\vartheta$ that are standard Brownian motions, the real part $X=e^B\cos \vartheta$ and imaginary part $Y=e^B\sin \vartheta$ are continuous local martingales. The Itô formula gives $dX=XdB-Yd\vartheta$ and $dY=YdB+Xd\vartheta$, since the two diagonal second-order terms cancel. Their quadratic variations equal ${\int }_0^t{e}^{2B_s}ds$ and their quadratic covariation is zero. If $C=[B,\vartheta ]$ is nonzero, the omitted drift terms are respectively $-YdC$ and $XdC$, so the orthogonality hypothesis is essential.

For continuous semimartingales and $Q\ll P$, their quadratic covariation has the same version under the two probability measures, up to $Q$-indistinguishability of stochastic processes. The dyadic product sums have the same limit by uniform convergence on compacts in probability under an absolutely continuous measure change. Uniqueness of a limit in probability identifies that limit with the bracket under $Q$. Semimartingale stability under an absolutely continuous measure change supplies the correct class under $Q$: a local martingale need not remain a local martingale. Taking the two processes equal proves the same statement for quadratic variation without a separate assumption.

If $Q\ll P$ and a continuous semimartingale is a semimartingale under both measures, its quadratic variations agree $Q$-indistinguishably. The same squared-increment sums converge uniformly on compacts in probability under both measures: absolute continuity transfers the original convergence, and uniqueness of the limit identifies the two continuous versions. Equivalence of measures is unnecessary.

For continuous local martingales $M$ and $N$, the Itô product rule says thatis a local martingale. If the martingales are square-integrable and converge in $L^2$, then

For continuous local martingales, the total-variation process of their quadratic covariation satisfiesIt is the Cauchy-Schwarz inequality for the matrix-valued measure formed by their quadratic variations and quadratic covariation.

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 sumsconverge 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 derivativesIf $(U,V)$ is a centered bivariate normal distribution with covariance matrix $\left(\begin{array}{cc}a& c\\ c& b\end{array}\right)$, thenLocalizing, 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 $|\mathbb{E}[UV]|\le \mathbb{E}|UV|\le \sqrt{\mathbb{E}{U}^2\mathbb{E}{V}^2}$,