Quadratic variation ($[X{]}_t$)

The quadratic variation of a continuous semimartingale is the limit in probabilityFinite-variation processes have zero quadratic variation, while a Brownian motion satisfies $[B{]}_t=t$.

For a zero-starting square-integrable martingale, the predictable quadratic variation is the predictable increasing process $A$ for which $M^2-A$ is a martingale. For a compensated jump process it integrates squared jump sizes against their rates. Unlike optional quadratic variation, it need not equal the sum of squared realized jumps. For continuous local martingales, the two brackets coincide after localization.

// Target: probability-and-statistics.bigb

For a continuous local martingale $M$, at most one continuous adapted increasing process $A$ starting at zero makes $M^2-A$ a local martingale. The difference of two candidates is a continuous finite-variation process and a local martingale, hence constant; its initial value is zero. Without the initial-value normalization only uniqueness up to an initial constant is possible.

Let $X$ be a continuous martingale on $[0,1]$ with $X_0=0$ and $|X_t|\le C$. For the dyadic squared-increment sums, including the unfinished increment at time $t$, set $A_t^{(n)}={\sum }_k(X_{k{2}^{-n}\wedge t}-X_{(k-1)2^{-n}\wedge t}{)}^2$. The associated martingale transform $M_t^{(n)}=(X_t^2-A_t^{(n)})/2$ satisfies $\mathbb{E}(M_1^{(n)}{)}^2\le C^4$ and $\mathbb{E}(A_1^{(n)}{)}^2\le 10C^4$. Discrete martingale orthogonality, the path modulus of continuity and the Cauchy-Schwarz inequality show that the terminal transforms are Cauchy in $L^2$. The Doob L2 maximal inequality then gives $\mathbb{E}{\sup }_{t\le 1}|A_t^{(n)}-A_t^{(m)}{|}^2\to 0$. The sums need not be increasing in time before taking the limit. This provides an elementary construction of quadratic variation under boundedness.

On a compact interval, squared increments of a continuous finite variation process are bounded by its modulus of continuity at the mesh size times its total variation, and therefore vanish. The Cauchy-Schwarz inequality bounds the mixed increment sum by the square root of the product of the two squared-increment sums. The local-martingale sum is bounded in probability, so the mixed term also vanishes.

Stop a continuous local martingale at increasing level-and-time stopping times to obtain bounded martingales. Their limits of discrete quadratic variation sums agree before the earlier stopping time, because the sums commute exactly with stopping. These continuous limits patch into a continuous adapted nondecreasing process. On each compact interval, the chance that the stopping time occurs early tends to zero, giving uniform convergence on compacts in probability for the original sums.

Completed squared increments form a nondecreasing step process. The continuous sum that includes the last partial increment differs by at most the square of the path's modulus of continuity over one mesh interval. Therefore both have the same limit in uniform convergence on compacts in probability. This proves monotonicity of the limiting quadratic variation without incorrectly asserting monotonicity of each continuous partial-increment sum.

For observations of an Itô integral on a partition of an interval, the statistic ${\sum }_{i}g(t_{i-1})(X_{t_i}-X_{t_{i-1}}{)}^2$ estimates $\int g(t)\sigma (t{)}^{2}dt$. With a deterministic noise coefficient the squared increments are independent, and the fluctuation has variance $2{\sum }_{i}g(t_{i-1}{)}^2({\int }_{t_{i-1}}^{t_i}{\sigma }^2{)}^2$.

The spot variance is the instantaneous squared noise coefficient $\sigma (t{)}^2$ of an Itô integral. It can be estimated by averaging the weighted realized variance over a shrinking time window. A Hölder continuity assumption bounds the smoothing bias.

A function with finite Hölder seminorm of exponent $\alpha >1/2$ has squared-increment sum on a uniform $N$-interval partition bounded by $C^2{N}^{1-2\alpha }$, hence tending to zero. This conflicts with a nonzero quadratic-variation limit. In particular dyadic quadratic variation of Brownian motion prevents a Brownian coordinate law supported on such a Hölder space. This argument alone gives no conclusion at the critical exponent $1/2$.

On $[0,1]$, the sum $Q_n$ of squared adjacent dyadic increments of standard Brownian motion has $\mathbb{E}{Q}_n=1$ and $\mathrm{var}(Q_n)=2^{1-n}$. Therefore $Q_n\to 1$ in $L^2$. On an interval of length $t$, scaling gives mean $t$ and variance $2t^2{2}^{-n}$. Independence is used within each grid, not between different grids.

Along the dyadic partitions of an interval of $[0,1]$, the sums ${\sum }_i|{\Delta }_{i}M{|}^p$ of a continuous local martingale tend to zero in probability for $p>2$. If $0<p<2$ and these sums are eventually bounded almost surely, then $M$ is constant on $[0,1]$: compare them with the sums defining quadratic variation, using the maximum increment, which tends to zero by uniform continuity.

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 $|\mathbb{E}[UV]|\le \mathbb{E}|UV|\le \sqrt{\mathbb{E}{U}^2\mathbb{E}{V}^2}$,

On $[0,T]$, a standard Brownian motion has pathwise quadratic variation $t$, whereas the time-rescaled process $B_{ct}$ has quadratic variation $ct$. Therefore their path laws are mutually singular when $c\ne 1$.