Worked Diagnostics for Quadratic Variation and Covariation

A Diagnostic Framework for Partition Sums

Quadratic variation and covariation problems often involve simple operations—scaling a Brownian motion, combining correlated coordinates, or adding a finite-variation path. The diagnostic work is to track what each operation does to the partition sums, calculate the relevant limit in the stated mode of convergence, and avoid confusing that limit with total variation.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying standard Brownian motions \(B_t\) and, when needed, \(C_t\). Assume \(C_t\) is independent of \(B_t\). No stochastic integrals are used in this tutorial. For \(T>0\), let \(\Pi=\{0=t_0<t_1<\cdots<t_n=T\}\) be a deterministic partition, let \(|\Pi|=\max_i(t_i-t_{i-1})\), and write \(\Delta_i X=X_{t_i}-X_{t_{i-1}}\).

Earlier material on Brownian quadratic variation establishes that \(Q_\Pi(B)\to T\) in mean square for deterministic partitions whose mesh tends to zero. For independent Brownian motions, their covariation sums tend to zero in mean square. These are convergence statements about sums over increasingly fine partitions; they do not assert that an individual squared increment equals its time increment.

Scaling Changes Quadratic Variation Quadratically

Proof. For every partition, \(\Delta_iX=a\Delta_iB\), so \(Q_\Pi(X)=a^2Q_\Pi(B)\). Therefore \(\mathbb{E}[(Q_\Pi(X)-a^2T)^2]=a^4\mathbb{E}[(Q_\Pi(B)-T)^2]\), which tends to zero by Brownian quadratic-variation convergence. This computation also covers \(a=0\), when the left-hand side is identically zero. \(\square\)

Correlated Coordinates and Covariation

Suppose \(B\) and \(C\) are independent standard Brownian motions and \(\rho\in[-1,1]\). The process \(D_t=\rho B_t+\sqrt{1-\rho^2}\,C_t\) is a standard Brownian motion correlated with \(B\), with covariation rate \(\rho\). At \(\rho=1\) or \(\rho=-1\), the coefficient of \(C\) is zero; the formulas below include these cases.

Proof. Write \(h_i=t_i-t_{i-1}\). Brownian increments on disjoint partition intervals are independent. For a centered normal variable with variance \(h_i\), its second moment is \(h_i\) and its fourth moment is \(3h_i^2\). Thus \(\mathbb{E}[Q_\Pi(B)]=T\) and \(\operatorname{Var}(Q_\Pi(B))=2\sum_i h_i^2\). The same formulas hold for \(C\).

For \(R_\Pi=\sum_i\Delta_iB\,\Delta_iC\), independence gives \(\mathbb{E}[R_\Pi]=0\). Its summands are independent across intervals, and each has variance \(\mathbb{E}[(\Delta_iB)^2]\mathbb{E}[(\Delta_iC)^2]=h_i^2\). Therefore \(\mathbb{E}[R_\Pi^2]=\sum_i h_i^2\). Since \(\sum_i h_i^2\le|\Pi|\sum_i h_i=T|\Pi|\), these variances tend to zero.

Now expand the sums directly: \(Q_\Pi(X)=a^2Q_\Pi(B)\), \(Q_\Pi(Y)=b^2\big(\rho^2Q_\Pi(B)+(1-\rho^2)Q_\Pi(C)+2\rho\sqrt{1-\rho^2}R_\Pi\big)\), and \(C_\Pi(X,Y)=ab\big(\rho Q_\Pi(B)+\sqrt{1-\rho^2}R_\Pi\big)\). The mean-square limits of \(Q_\Pi(B)\), \(Q_\Pi(C)\), and \(R_\Pi\) just proved give the three stated limits. For \(a=0\) or \(b=0\), the corresponding sums vanish identically, consistent with the formulas. \(\square\)

Finite Variation: Zero Quadratic Variation, Not Zero Total Variation

Proof. The total variation \(V_T(A)\) is the supremum, over partitions of \([0,T]\), of the sum of absolute increments of \(A\); assume it is finite. For each partition, let \(m_\Pi=\max_i|\Delta_iA|\). Continuity on the compact interval implies uniform continuity, so \(m_\Pi\to0\) as \(|\Pi|\to0\). Consequently, \(Q_\Pi(A)\le m_\Pi\sum_i|\Delta_iA|\le m_\Pi V_T(A)\to0\).

The finite-sum Cauchy–Schwarz inequality gives \(|C_\Pi(A,B)|^2\le Q_\Pi(A)Q_\Pi(B)\). To verify convergence in probability, fix \(\varepsilon>0\) and \(\delta>0\). Brownian quadratic variation converges in probability to \(T\), so for sufficiently fine partitions \(\mathbb{P}(Q_\Pi(B)>T+1)<\delta\). On the event \(Q_\Pi(B)\le T+1\), the bound on \(Q_\Pi(A)\) tends to zero almost surely. Hence the probability that \(|C_\Pi(A,B)|>\varepsilon\) has limit superior at most \(\delta\). Since \(\delta\) was arbitrary, this probability tends to zero. \(\square\)

Convergence Modes and Common Errors

Mean-square convergence implies convergence in probability by Chebyshev's inequality: if \(\mathbb{E}[(Z_n-z)^2]\to0\), then for every \(\varepsilon>0\), \(\mathbb{P}(|Z_n-z|>\varepsilon)\le \mathbb{E}[(Z_n-z)^2]/\varepsilon^2\to0\). Almost-sure convergence is a separate assertion. Convergence in probability alone does not imply almost-sure convergence along the same sequence; almost-sure claims require a suitable argument, such as a summable probability bound and Borel–Cantelli. Keep the mode supplied by the calculation rather than upgrading it without justification.

Common mistake—scaling only once. A wrong calculation might say \(Q_\Pi(2B)=2Q_\Pi(B)\), and hence claim a limit \(2T\). But each increment is \(2\Delta_iB\), so its square is \(4(\Delta_iB)^2\); summing gives \(Q_\Pi(2B)=4Q_\Pi(B)\), with limit \(4T\). For example, at \(T=2\) the correct limit is \(4\cdot2=8\), not \(2\cdot2=4\).

Another error is to treat quadratic variation as total variation. For \(A_t=t^2\), the total variation on \([0,1]\) is \(1\), whereas the quadratic-variation sums tend to zero. Conversely, Brownian motion has quadratic variation \(T\) and almost surely infinite total variation. These quantities answer different questions: quadratic variation tracks sums of squared increments; total variation tracks sums of absolute increments.