Gaussian Moments of Brownian Increment Squares

Why the Squares Have Nonzero Variance

Brownian increments are centered, but their squares are not deterministic: an increment over a time interval of length \(h\) has a normal distribution with variance \(h\), and its square has mean \(h\) and variance \(2h^2\). For a partition, the exact variance of the sum of squared increments therefore depends on the lengths of the individual intervals, not just on their total length. Cross-moments also depend on whether the intervals overlap.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\) relative to this filtration. In particular, \(B_t-B_s\) is independent of \(\mathcal{F}_s\) and has distribution \(N(0,t-s)\) whenever \(0\le s<t\). For an interval \([s,t]\), write \(\Delta_{s,t}B=B_t-B_s\). No stochastic integral is needed here.

Moments of One Increment

Let \(Z\sim N(0,h)\), where \(h\ge0\). Its mean is zero and its second moment is \(h\). For \(h>0\), its density is \(\varphi_h(x)=(2\pi h)^{-1/2}e^{-x^2/(2h)}\). Since \(\varphi_h'(x)=-(x/h)\varphi_h(x)\), integration by parts gives \(\int_{\mathbb{R}}x^{2k}\varphi_h(x)\,dx=(2k-1)h\int_{\mathbb{R}}x^{2k-2}\varphi_h(x)\,dx\) for \(k\ge1\). The boundary term vanishes because a polynomial times the Gaussian density tends to zero at both ends. Taking \(k=1\) and \(k=2\) yields the second and fourth moments below. If \(h=0\), then \(Z=0\) almost surely and the same formulas hold.

Proof. For \(h>0\), the integration-by-parts recurrence gives \(\mathbb{E}[Z^2]=h\mathbb{E}[1]=h\), then \(\mathbb{E}[Z^4]=3h\mathbb{E}[Z^2]=3h^2\). Thus \(\operatorname{Var}(Z^2)=\mathbb{E}[Z^4]-(\mathbb{E}[Z^2])^2=3h^2-h^2=2h^2\). The case \(h=0\) follows from \(Z=0\) almost surely. For the conditional identities, \(B_t-B_s\) is independent of \(\mathcal{F}_s\) and has distribution \(N(0,t-s)\). Hence its conditional moments equal its unconditional moments just computed. \(\square\)

The conditional identities are useful because the moment values do not depend on the information available at the starting time. They do not assert that the squared increment itself equals the interval length; they give its conditional mean. The fourth moment quantifies how much the square fluctuates around that mean.

Cross-Moments and Overlap

Two increments on disjoint time intervals have independent values, so their squares are independent as well. When intervals overlap, the squares generally are not independent. The overlap determines the covariance of the increments, and that covariance determines the cross-moment of their squares.

Proof. First, \(\operatorname{Cov}(X,Y)=c\). To verify this, use \(\mathbb{E}[B_rB_q]=\min(r,q)\), which follows by writing the Brownian values as their shared increment up to the earlier time plus an independent later increment. Expanding the covariance gives \(\min(t,v)-\min(t,u)-\min(s,v)+\min(s,u)\), which is the length of the intersection of \([s,t]\) and \([u,v]\), namely \(c\). The increment variances are \(a\) and \(b\).

If \(a=0\), then \(X=0\) almost surely and \(c=0\), so both sides of the cross-moment formula are zero. Suppose \(a>0\), and put \(R=Y-(c/a)X\). The pair \((X,Y)\) is jointly Gaussian, so \(X,R\) are jointly Gaussian; moreover \(\operatorname{Cov}(X,R)=c-(c/a)a=0\), and jointly Gaussian variables with zero covariance are independent. Set \(r=\mathbb{E}[R^2]=b-c^2/a\), which is nonnegative because it is a variance. Now \(Y=(c/a)X+R\). Independence, \(\mathbb{E}[R]=\mathbb{E}[X]=0\), and the vanishing odd moments of centered normal variables give \(\mathbb{E}[X^2Y^2]=(c/a)^2\mathbb{E}[X^4]+\mathbb{E}[X^2]\mathbb{E}[R^2]=3c^2+ar=3c^2+ab-c^2=ab+2c^2\). This also covers \(r=0\), when \(R=0\) almost surely. Subtracting \(\mathbb{E}[X^2]\mathbb{E}[Y^2]=ab\) gives \(\operatorname{Cov}(X^2,Y^2)=2c^2\). \(\square\)

In particular, disjoint intervals have \(c=0\), and the cross-moment factors as \(ab\). For identical intervals \(c=a=b\), the formula gives \(3a^2\), the fourth moment of a single increment. The formula thus includes both independent and fully coincident increments, as well as partial overlap.

Exact Moments of Partition Sums

Take a deterministic partition \(\Pi=\{0=t_0<t_1<\cdots<t_n=T\}\), and let \(h_i=t_i-t_{i-1}\). Its partition-based quadratic-variation sum is \(Q_\Pi=\sum_{i=1}^n(B_{t_i}-B_{t_{i-1}})^2\). The increments on these consecutive intervals are independent, by the independent-increments property of Brownian motion. Their squares are therefore independent too.

Proof. Write \(X_i=B_{t_i}-B_{t_{i-1}}\), so \(X_i\sim N(0,h_i)\). Linearity of expectation and the one-increment moment formula give \(\mathbb{E}[Q_\Pi]=\sum_i\mathbb{E}[X_i^2]=\sum_i h_i=T\). Independence of the \(X_i^2\) gives \(\operatorname{Var}(Q_\Pi)=\sum_i\operatorname{Var}(X_i^2)=2\sum_i h_i^2\). Finally, the defining identity \(\operatorname{Var}(Q_\Pi)=\mathbb{E}[Q_\Pi^2]-(\mathbb{E}[Q_\Pi])^2\) rearranges to \(\mathbb{E}[Q_\Pi^2]=T^2+2\sum_i h_i^2\). \(\square\)

A useful extension follows from the same computation. For fixed real weights \(\lambda_1,\ldots,\lambda_n\), define \(R=\sum_i\lambda_iX_i^2\). Linearity gives \(\mathbb{E}[R]=\sum_i\lambda_i h_i\). Independence gives \(\operatorname{Var}(R)=\sum_i\lambda_i^2\operatorname{Var}(X_i^2)=2\sum_i\lambda_i^2h_i^2\). The weights need not be positive; the variance formula uses their squares.

A Common Variance Pitfall

Do not replace the variance of a partition sum by the variance of the square of the total increment. For example, with four equal intervals on \([0,1]\), each length is \(1/4\). The correct variance is \(2\cdot4(1/4)^2=2\cdot4\cdot1/16=1/2\). A tempting but wrong calculation treats the sum as \(B_1^2\), giving \(\operatorname{Var}(B_1^2)=2\). But \(Q_\Pi\) is a sum of four squared increments, not the square of their sum: expanding \((\sum_iX_i)^2\) also introduces cross terms \(2X_iX_j\).

Check: the correct mean is \(4(1/4)=1\), and its second moment is \(1^2+1/2=3/2\). The incorrect value \(2\) would instead give a second moment \(1^2+2=3\), inconsistent with the direct sum of squared-increment moments. Exact finite-partition moments provide a useful diagnostic: each interval contributes \(2h_i^2\) to the variance, while overlap contributes \(2c^2\) to the covariance of two squared increments.