Multidimensional SDEs and Correlated Noise

From Scalar Noise to Vector Noise

A scalar SDE has one state variable and one Brownian driver. In a multidimensional model, the number of state variables and the number of noise sources need not agree. One Brownian coordinate may affect several state variables, and a state variable may receive noise from several coordinates. The diffusion matrix records these loadings; its product with its transpose gives the instantaneous covariance structure of the stochastic part.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})$ carrying a standard $m$-dimensional Brownian motion $\mathbf{B}_t=(B_t^1,\ldots,B_t^m)^\mathsf{T}$. Its coordinates are independent standard Brownian motions relative to the filtration. Let the state $X_t$ take values in $\mathbb{R}^n$. The drift $b(t,X_t)$ is an $n$-vector, and the diffusion coefficient $\sigma(t,X_t)$ is an $n\times m$ matrix.

Definition: A vector SDE driven by standard $m$-dimensional Brownian motion is written $dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\,d\mathbf{B}_t$, or equivalently as its integral equation $X_t=X_0+\int_0^t b(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,d\mathbf{B}_s$. The $i$th coordinate of the stochastic integral is $\sum_{k=1}^m\int_0^t\sigma_{ik}(s,X_s)\,dB_s^k$. Each integral is an Ito integral: its integrand is progressively measurable and belongs to $\mathcal{L}^2$, meaning $\mathbb{E}\int_0^T\sum_{k=1}^m|\sigma_{ik}(s,X_s)|^2\,ds<\infty$ for every finite $T$. A sufficient drift condition for the displayed ordinary integral is $\int_0^T|b(s,X_s)|\,ds<\infty$ almost surely.

The dimensions are part of the formulation: $b$ has $n$ entries, $\sigma$ has $n$ rows and $m$ columns, and $d\mathbf{B}_t$ has $m$ entries. Thus the matrix product $\sigma\,d\mathbf{B}_t$ is an $n$-vector. The drift describes the finite-variation rate; the diffusion describes how each Brownian coordinate enters the state.

The Diffusion Matrix and Covariation

For row $i$ of $\sigma$, the $i$th stochastic differential is $\sum_k\sigma_{ik}\,dB_t^k$. Since the Brownian coordinates are independent, their multiplication rules are $dB_t^k\,dB_t^\ell=0$ for $k\ne\ell$, $dB_t^k\,dB_t^k=dt$, and $dB_t^k\,dt=dt\,dt=0$. It follows that the matrix of instantaneous covariation rates is $\sigma\sigma^\mathsf{T}$, an $n\times n$ matrix. This is not generally the covariance matrix of $X_t$ itself: the latter concerns the accumulated, possibly state-dependent evolution over time.

Theorem: Suppose $X$ satisfies the vector SDE above on $[0,T]$, with progressively measurable diffusion entries in $\mathcal{L}^2$. Write $M_t=\int_0^t\sigma(s,X_s)\,d\mathbf{B}_s$ for its stochastic part. Then, for $1\le i,j\le n$, its quadratic covariation is $$ [M_i,M_j]_t=\int_0^t\sum_{k=1}^m\sigma_{ik}(s,X_s)\sigma_{jk}(s,X_s)\,ds. $$ In matrix form the rate of this covariation is $a(t,X_t)=\sigma(t,X_t)\sigma(t,X_t)^\mathsf{T}$, called the instantaneous covariance matrix.

Proof. Each coordinate $M_i=\sum_k\int_0^\cdot\sigma_{ik}(s,X_s)\,dB_s^k$ is a square-integrable martingale: the Ito isometry gives $\mathbb{E}[M_i(t)^2]=\mathbb{E}\int_0^t\sum_k\sigma_{ik}(s,X_s)^2\,ds<\infty$, and the martingale property follows from the construction of the Ito integral. In particular, the defining conditional-mean property follows first for simple adapted integrands from the zero conditional means of Brownian increments, and then for $\mathcal{L}^2$ integrands by approximation and continuity of conditional expectation in $L^1$. Apply the Ito product formula to $M_iM_j$. Its finite-variation term is $\sum_k\sigma_{ik}\sigma_{jk}\,dt$: terms involving distinct Brownian coordinates vanish, and each equal-coordinate product contributes $dt$. By the definition of quadratic covariation as the finite-variation term in this product formula, integrating from $0$ to $t$ gives the stated expression. $\square$

The matrix $a=\sigma\sigma^\mathsf{T}$ is symmetric and positive semidefinite: for every $v\in\mathbb{R}^n$, $v^\mathsf{T}av=|\sigma^\mathsf{T}v|^2\ge0$. It need not be invertible. If the rows of $\sigma$ are linearly dependent, some nonzero linear combination of the state coordinates has no instantaneous stochastic variation.

Worked Example: Two State Variables with Independent Noise

On a finite time interval, let $B^1,B^2$ be independent Brownian coordinates and consider $dX_t^1=dt+2\,dB_t^1$, $dX_t^2=-dt+3\,dB_t^2$, with deterministic initial values. The constant integrands are progressively measurable and square-integrable on every finite interval, so both Ito integrals are defined.

Here $n=m=2$, $b=(1,-1)^\mathsf{T}$, and $\sigma=\begin{pmatrix}2&0\\0&3\end{pmatrix}$. Multiplying the matrices gives

a=\sigma\sigma^\mathsf{T} =\begin{pmatrix}2&0\\0&3\end{pmatrix} \begin{pmatrix}2&0\\0&3\end{pmatrix} =\begin{pmatrix}4&0\\0&9\end{pmatrix}.

Thus $[M_1]_t=4t$, $[M_2]_t=9t$, and $[M_1,M_2]_t=0$. For a direct check, $M_1(t)=2B_t^1$ and $M_2(t)=3B_t^2$; their individual quadratic variations are $4t$ and $9t$, while independent Brownian coordinates have zero covariation. The drift entries do not contribute to quadratic covariation.

Correlated Brownian Drivers

Models are often described using Brownian inputs that are correlated rather than independent. A convenient way to formulate them is to express the correlated inputs as linear combinations of independent Brownian coordinates. Let $R$ be a symmetric positive semidefinite $q\times q$ matrix with diagonal entries equal to one. If $L$ is a $q\times m$ matrix satisfying $LL^\mathsf{T}=R$, define $\mathbf{W}_t=L\mathbf{B}_t$, where $\mathbf{B}$ is standard $m$-dimensional Brownian motion. Then $\mathbf{W}$ has continuous paths, centered Gaussian increments, independent increments over disjoint time intervals, and increment covariance $(t-s)R$ over $[s,t]$. Its coordinates are standard Brownian motions individually, with correlations given by the off-diagonal entries of $R$.

Proposition: Suppose a model is written $dX_t=b(t,X_t)\,dt+G(t,X_t)\,d\mathbf{W}_t$, where $\mathbf{W}=L\mathbf{B}$ and $LL^\mathsf{T}=R$. Then the same stochastic term, expressed using the independent standard Brownian coordinates $\mathbf{B}$, has diffusion matrix $\sigma=GL$. Its instantaneous covariance matrix is $GRG^\mathsf{T}$.

Proof. Since $\mathbf{W}=L\mathbf{B}$, the stochastic term is $G\,d\mathbf{W}=GL\,d\mathbf{B}$, so its diffusion matrix relative to $\mathbf{B}$ is $\sigma=GL$. The covariance formula for independent Brownian coordinates gives $\sigma\sigma^\mathsf{T}$. Substitution and associativity of matrix multiplication yield $(GL)(GL)^\mathsf{T}=GLL^\mathsf{T}G^\mathsf{T}=GRG^\mathsf{T}$. This also shows why the covariance depends on both the loading matrix $G$ and the correlations $R$. $\square$

Every symmetric positive semidefinite matrix $R$ admits such a factorization, for example by the spectral theorem. If $R$ is singular, $L$ can still be chosen, but the correlated coordinates have a linear dependence and the number of independent Brownian coordinates needed may be smaller than $q$. A matrix with a negative eigenvalue cannot be a covariance matrix, so not every proposed table of correlations defines a valid noise model.

Worked Example: Two Correlated Inputs

Let $B^1,B^2$ be independent Brownian coordinates and define $W^1=B^1$, $W^2=\tfrac35B^1+\tfrac45B^2$. These are standard Brownian motions individually. Their covariance matrix per unit time is $R=\begin{pmatrix}1&3/5\\3/5&1\end{pmatrix}$: the off-diagonal entry follows from the shared $B^1$ term, and the variance of $W^2$ per unit time is $(3/5)^2+(4/5)^2=9/25+16/25=1$.

Consider $dX_t^1=2\,dW_t^1$, $dX_t^2=3\,dW_t^2$, with deterministic initial state. All integrands are constant and belong to $\mathcal{L}^2$ on finite intervals. In terms of independent drivers,

\sigma=\begin{pmatrix}2&0\\[2pt]9/5&12/5\end{pmatrix}, \qquad a=\sigma\sigma^\mathsf{T} =\begin{pmatrix}4&18/5\\18/5&9\end{pmatrix}.

For the second diagonal entry, $(9/5)^2+(12/5)^2=81/25+144/25=225/25=9$. The off-diagonal entry is $2(9/5)+0(12/5)=18/5$. Equivalently, using the correlated-driver representation, $[M_1,M_2]_t=2\cdot3\cdot(3/5)t=18t/5$; the two calculations agree. The positive sign records the positive correlation of the input noises.

Worked Example: Mixing Correlated Noise Across State Coordinates

Take correlated inputs with $W^1=B^1$, $W^2=-\tfrac12B^1+\tfrac{\sqrt3}{2}B^2$. Their covariance matrix per unit time is $R=\begin{pmatrix}1&-1/2\\-1/2&1\end{pmatrix}$, since $\operatorname{Var}(W^2)$ per unit time is $1/4+3/4=1$, and the shared-coordinate covariance is $-1/2$. Let the stochastic loadings of two state coordinates be $dX_t^1=2\,dW_t^1+dW_t^2$ and $dX_t^2=-dW_t^1+2\,dW_t^2$. These constant integrands are square-integrable on finite intervals.

The loading matrix relative to $\mathbf{W}$ is $G=\begin{pmatrix}2&1\\-1&2\end{pmatrix}$. Replacing the correlated inputs by independent ones gives

\sigma=GL =\begin{pmatrix}2&1\\-1&2\end{pmatrix} \begin{pmatrix}1&0\\-1/2&\sqrt3/2\end{pmatrix} =\begin{pmatrix}3/2&\sqrt3/2\\-2&\sqrt3\end{pmatrix}.

Consequently, $a=\sigma\sigma^\mathsf{T}=\begin{pmatrix}3&-3/2\\-3/2&7\end{pmatrix}$. Indeed, the first diagonal is $(3/2)^2+(\sqrt3/2)^2=9/4+3/4=3$; the second is $(-2)^2+(\sqrt3)^2=4+3=7$; and the cross-entry is $(3/2)(-2)+(\sqrt3/2)(\sqrt3)=-3+3/2=-3/2$. Directly from the correlated drivers, the first variance rate is $4+1+2(2)(1)(-1/2)=3$; the second is $1+4+2(-1)(2)(-1/2)=7$; and the cross-rate is $-2-2+1/2+2=-3/2$. The two representations give the same covariance matrix.

What the Covariance Matrix Does—and Does Not—Say

The matrix $a(t,X_t)$ describes local stochastic variation, not the full distribution of the state. In particular, nonzero off-diagonal entries mean the stochastic parts of two coordinates have nonzero instantaneous covariation. A zero entry means their instantaneous covariation rate is zero; it does not by itself imply that the coordinates of $X_t$ are independent. Drift, dependence through state-dependent coefficients, and their common history can all affect the joint distribution.

Nor does a diffusion matrix have to be square. With $n$ state variables and $m$ independent noise sources, it is $n\times m$. Its rank determines how many independent directions of instantaneous stochastic variation are present, while $a=\sigma\sigma^\mathsf{T}$ is always $n\times n$. Different diffusion matrices can produce the same $a$, so the covariance matrix alone need not identify the particular noise representation.

Common Mistake: Forgetting Cross Terms

A frequent error is to calculate each row's variance correctly but omit the products between rows when computing covariation. For example, in the preceding mixed-noise calculation, a mistaken calculation might give the cross-rate as $(3/2)(-2)=-3$, using only the first independent Brownian coordinate. The second independent coordinate contributes $(\sqrt3/2)(\sqrt3)=3/2$, so the correct cross-rate is $-3+3/2=-3/2$. The reliable rule is to multiply corresponding entries across the two rows and sum over all independent Brownian coordinates.

Also distinguish a covariance rate from an ordinary covariance. For constant $\sigma$ and zero drift, the stochastic increment over an interval of length $h$ has covariance $h\sigma\sigma^\mathsf{T}$. For state-dependent $\sigma$, the accumulated bracket is instead the random integral $\int_0^t\sigma(s,X_s)\sigma(s,X_s)^\mathsf{T}\,ds$. It should not be replaced by $t$ times a fixed matrix unless the relevant coefficients are constant.

Check Your Understanding

For each question, keep the number of state coordinates separate from the number of Brownian drivers.

If $X$ has dimension $3$ and is driven by $2$ independent Brownian coordinates, what are the dimensions of $\sigma$ and $\sigma\sigma^\mathsf{T}$?
For $\sigma=\begin{pmatrix}1&2\\3&-1\end{pmatrix}$, compute each entry of the instantaneous covariance matrix.
Let $W^1=B^1$ and $W^2=\rho B^1+\sqrt{1-\rho^2}B^2$, where $-1\le\rho\le1$. Compute their covariance matrix per unit time, including the endpoint cases $\rho=\pm1$.
Explain why a zero off-diagonal entry of $\sigma\sigma^\mathsf{T}$ does not, on its own, establish independence of the state coordinates.
For correlated input noise with covariance matrix $R$ and loading matrix $G$, state the independent-driver diffusion matrix and the resulting instantaneous covariance matrix.