Computing Brownian Covariance Matrices

From Observation Times to a Covariance Matrix

A finite collection of Brownian observations is a Gaussian vector, and its covariance matrix records how strongly each pair of observations varies together. For a specified time grid, the entries are determined by the earlier covariance formula for standard Brownian motion: the covariance at two times is the smaller time. The main computational work is therefore to place the times in the correct positions, handle irregular spacing without rounding, and check what the resulting matrix says about the observations.

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space carrying standard Brownian motion \((B_t)_{t\ge0}\). For observation times \(t_1,\ldots,t_n\), define the random vector \(Y=(B_{t_1},\ldots,B_{t_n})^\mathsf{T}\). The covariance matrix of \(Y\), denoted \(C\), has entry \(C_{ij}=\operatorname{Cov}(B_{t_i},B_{t_j})\). Standard Brownian motion has mean zero and covariance function \(\operatorname{Cov}(B_s,B_t)=\min(s,t)\), so each entry is obtained by taking the smaller of the two times.

The order of the times in the list determines the order of rows and columns, not the values of the covariances themselves. If the times are listed chronologically, the matrix has a useful structure: each row agrees with the earlier rows up through that observation's time. If the times are listed in another order, construct the entries in that listed order or sort the times first and then apply the same permutation to both rows and columns.

A Factorization for Ordered Times

For \(0\le t_1\le t_2\le\cdots\le t_n\), set \(t_0=0\), and define the gaps \(\Delta_k=t_k-t_{k-1}\) for \(1\le k\le n\). Let \(L\) be the lower-triangular matrix whose entry \(L_{ik}\) is \(1\) when \(k\le i\), and \(0\) otherwise. Then the covariance matrix factors into \(L\), a diagonal matrix of time gaps, and the transpose of \(L\).

Proof. For indices \(i,j\), the \((i,j)\) entry of the product is \(\sum_{k=1}^n L_{ik}\Delta_k L_{jk}\). The factors \(L_{ik}L_{jk}\) equal \(1\) exactly when \(k\le\min(i,j)\), and otherwise equal \(0\). Thus this entry is \(\sum_{k=1}^{\min(i,j)}(t_k-t_{k-1})=t_{\min(i,j)}=\min(t_i,t_j)=C_{ij}\), proving the factorization. For any column vector \(x\), \(x^\mathsf{T}Cx=(L^\mathsf{T}x)^\mathsf{T}\operatorname{diag}(\Delta_1,\ldots,\Delta_n)(L^\mathsf{T}x)=\sum_{k=1}^n\Delta_k((L^\mathsf{T}x)_k)^2\ge0\), since all gaps are nonnegative. This proves positive semidefiniteness. The matrix \(L\) is triangular with every diagonal entry \(1\), so \(\det(L)=\det(L^\mathsf{T})=1\). The determinant product rule now gives \(\det(C)=\prod_k\Delta_k\). If all gaps are positive, then for every nonzero \(x\), \(L^\mathsf{T}x\ne0\) because \(L^\mathsf{T}\) is invertible, so the displayed sum is strictly positive; hence \(C\) is positive definite. If some gap is zero, the determinant is zero, so \(C\) is not positive definite. The first gap is positive exactly when \(t_1>0\), and each later gap is positive exactly when the times strictly increase. This proves the claimed condition. \(\square\)

This factorization is also a practical check on a calculation. The determinant depends on the successive gaps, not on a product of the observation times. A zero first time or a repeated time creates a zero gap and makes the matrix singular: \(B_0=0\) with probability one, and two observations at the same time are identical random variables.

Unsorted Lists and Singular Grids

Observation times need not be presented in chronological order. The entry formula still applies directly. For determinant or definiteness calculations, sorting is often simpler: if a permutation reorders the times, it applies the same permutation to rows and columns. Such a simultaneous reordering does not change the determinant, since the two permutation determinants multiply to \(1\).

If the times are nonnegative but not all distinct and positive, the same entrywise definition still gives a valid covariance matrix; only strict positive definiteness can fail. The gap factorization proves positive semidefiniteness even in these cases. For repeated times, the corresponding rows and columns agree. For a time equal to zero, its row and column consist entirely of zeros.

Why the Matrix Matters—and a Common Error

The covariance matrix is the full second-order description of the finite vector \(Y\). In particular, the diagonal reports each observation's variance, and the off-diagonal entries report pairwise covariances. Because the finite-dimensional laws of Brownian motion are Gaussian, the mean vector together with this covariance matrix specifies the joint Gaussian law. Thus an incorrectly entered off-diagonal entry changes the joint law, not merely the appearance of a table.

Common mistake: using the larger time. For observations at times \(\tfrac12\) and \(\tfrac32\), a wrong calculation might set their covariance to \(\tfrac32\), the larger time. The correction is \(\min(\tfrac12,\tfrac32)=\tfrac12\). Using \(\tfrac32\) would even violate the covariance bound \(|\operatorname{Cov}(U,V)|\le\sqrt{\operatorname{Var}(U)\operatorname{Var}(V)}\): here the bound is \(\sqrt{(\tfrac12)(\tfrac32)}=\tfrac{\sqrt3}{2}\), which is less than \(\tfrac32\). A reliable workflow is to write the time list first, fill every entry using the minimum, check symmetry and the diagonal, and then use sorted time gaps to check the determinant and definiteness.