Quadratic Variation and Total Variation in Worked Examples

Two Different Ways to Measure Path Movement

For a continuous path \(x:[0,T]\to\mathbb{R}\), a partition records its increments but does not, by itself, determine how the path moved between partition points. The sum of absolute increments measures movement visible on that partition. The sum of squared increments gives greater weight to large increments and, for Brownian motion, has a nonzero limit under refinement even though the path’s absolute-increment sums diverge.

We work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). For a continuous path \(x\) and a partition \(\Pi=\{0=t_0<t_1<\cdots<t_n=T\}\), write \(\Delta_i x=x(t_i)-x(t_{i-1})\), and let \(|\Pi|=\max_i(t_i-t_{i-1})\) be its mesh.

Refining a partition cannot decrease its absolute-increment sum: if an increment is split into several increments, the triangle inequality says that the sum of their absolute values is at least the absolute value of the original increment. Squared-increment sums do not have that monotonicity. A partition that misses a change of direction can therefore record little or no movement in either sum, even when the path moves substantially between its endpoints.

Linear Paths: Absolute Movement Persists, Squares Vanish

The linear calculation reflects a general finite-variation estimate. For a continuous path, let \(m_\Pi=\max_i|\Delta_i x|\). Then \(Q_\Pi(x)\le m_\Pi A_\Pi(x)\). As the mesh tends to zero, continuity makes \(m_\Pi\) tend to zero. If the path has finite total variation, \(A_\Pi(x)\) is bounded by that total variation for every partition, so the squared-increment sums must vanish.

Proof. Write \(V=\operatorname{TV}_{[0,T]}(x)<\infty\). For every partition, \(|\Delta_i x|\le m_\Pi\), so \((\Delta_i x)^2\le m_\Pi|\Delta_i x|\). Summing gives \(0\le Q_\Pi(x)\le m_\Pi A_\Pi(x)\le m_\Pi V\). A continuous function on the compact interval \([0,T]\) is uniformly continuous; therefore the largest increment \(m_\Pi\) tends to zero whenever \(|\Pi|\to0\). The displayed bound then gives \(Q_\Pi(x)\to0\). This also covers a constant path, for which \(V=m_\Pi=Q_\Pi(x)=0\). \(\square\)

Piecewise-Linear Paths: A Partition Can Hide Movement

Proof. Let the path have \(r\) interior breakpoints and let \(L\) be the largest absolute value of its finitely many slopes. Refine an arbitrary partition by adding all the breakpoints. On each resulting interval the path is linear, so the sum of absolute increments on the refined partition equals the sum of the absolute changes along all linear pieces; this is precisely the total variation \(V\). The triangle inequality gives \(A_\Pi(x)\le V\). Only original partition intervals containing an interior breakpoint can contribute a difference between the refined and original absolute-increment sums. There are at most \(r\) such intervals. Each has length at most \(|\Pi|\), and the total variation within it is at most \(L|\Pi|\). Thus \(0\le V-A_\Pi(x)\le rL|\Pi|\), which tends to zero. Finally, \(|\Delta_i x|\le L|\Pi|\) and \(A_\Pi(x)\le V\), so \(Q_\Pi(x)\le L|\Pi|V\to0\). If \(L=0\) or \(r=0\), the same inequalities still apply; a constant path has both sums zero. \(\square\)

The coarse partition in the example demonstrates why a single absolute-increment sum is not total variation. A path can return to its starting point, causing cancellation in the endpoint increment. Refining so that the turning point is recorded reveals the movement. For the piecewise-linear path, the proposition ensures that all sufficiently fine partitions capture the total variation, even if their points do not land exactly on the corners.

Brownian Paths: Finite Quadratic Variation, Infinite Total Variation

For Brownian motion, the two sums behave differently from those of a continuous finite-variation path. For any deterministic partition of \([0,T]\), put \(h_i=t_i-t_{i-1}\). The increments are independent and distributed as \(N(0,h_i)\). A centered normal variable with variance \(h_i\) has second moment \(h_i\), fourth moment \(3h_i^2\), and absolute first moment \(\sqrt{2h_i/\pi}\). These facts give both the mean-square behavior of the squared-increment sum and the expected absolute-increment sum.

Proof. For a deterministic partition, independence and the normal fourth-moment calculation give \(\mathbb{E}[Q_\Pi(B)]=\sum_i h_i=T\) and \(\operatorname{Var}(Q_\Pi(B))=2\sum_i h_i^2\le2T|\Pi|\). Hence \(\mathbb{E}[(Q_\Pi(B)-T)^2]\le2T|\Pi|\to0\), proving the \(L^2\) assertion. For the dyadic partitions \(\Pi_k\), the mesh is \(T2^{-k}\), so the variance bound is \(2T^2 2^{-k}\). For every \(\varepsilon>0\), Chebyshev’s inequality gives \(\mathbb{P}(|Q_{\Pi_k}(B)-T|>\varepsilon)\le 2T^2 2^{-k}/\varepsilon^2\); the right-hand side is summable in \(k\). The Borel–Cantelli lemma, applied for each positive rational \(\varepsilon\), yields \(Q_{\Pi_k}(B)\to T\) almost surely. On an outcome where this convergence holds and the Brownian path is continuous, let \(m_k=\max_i|\Delta_i B|\) on \(\Pi_k\). Uniform continuity gives \(m_k\to0\), and \(Q_{\Pi_k}(B)\le m_k A_{\Pi_k}(B)\). Since \(T>0\), eventually \(Q_{\Pi_k}(B)>T/2\); then \(A_{\Pi_k}(B)\ge Q_{\Pi_k}(B)/m_k\to\infty\). This proves the almost-sure divergence along dyadic partitions. \(\square\)

The modes of convergence in this result matter. Along arbitrary deterministic partitions whose mesh tends to zero, the squared sums converge to \(T\) in \(L^2\), and therefore in probability. The proof gives almost-sure convergence along dyadic partitions; it does not assert almost-sure convergence along every deterministic sequence of refining partitions. The dyadic absolute sums diverge almost surely, so Brownian motion has infinite total variation almost surely. This does not conflict with its finite quadratic variation: quadratic variation is not accumulated absolute path movement.

Common Mistakes and a Practical Comparison

Mistake: treating endpoint change as total movement. For the triangular path, using only the endpoints gives \(|x(1)-x(0)|=0\) and may lead to the incorrect claim that its total variation is zero. The correction is to include the turning point: \(|x(1/2)-x(0)|+|x(1)-x(1/2)|=1/2+1/2=1\). Total variation is a supremum over partitions, not the absolute endpoint change.

Mistake: calling the Brownian squared-sum limit a pathwise statement for every refining sequence. The calculation \(\mathbb{E}[(Q_\Pi(B)-T)^2]\le2T|\Pi|\) establishes \(L^2\) convergence, not almost-sure convergence for every sequence. The almost-sure claim proved here is specifically along dyadic partitions. This distinction is essential when comparing a mode of convergence with a pathwise total-variation statement.

The comparison is useful whenever a partition calculation is used to infer a property of an entire path. For smooth or piecewise-smooth motion, absolute increments recover accumulated movement under refinement, while the squared sums vanish. Brownian motion has the opposite signature: its squared increments retain a finite limit, while its absolute increments reveal unbounded total movement. Neither sum should be substituted for the other.