Drift and Diffusion Coefficients: Domains and Dimensions

State Space and Noise Space Are Different

An SDE has two distinct dimensions to keep track of. The state dimension is the number of coordinates of the process being modelled. The noise dimension is the number of Brownian coordinates driving it. These dimensions need not be equal: a scalar process can be driven by several Brownian motions, and a vector process can be driven by a single Brownian motion.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard \(m\)-dimensional Brownian motion \(\mathbf{B}_t=(B_t^1,\ldots,B_t^m)^\mathsf{T}\) relative to the filtration. Fix a finite time horizon \(T>0\). Let the state take values in a set \(D\subseteq\mathbb{R}^n\). The state dimension is \(n\); the noise dimension is \(m\).

Definition: For a state process \(X_t\in D\subseteq\mathbb{R}^n\) driven by \(m\)-dimensional Brownian motion, the drift coefficient has the form \(b:[0,T]\times D\to\mathbb{R}^n\), and the diffusion coefficient has the form \(\sigma:[0,T]\times D\to\mathbb{R}^{n\times m}\). The corresponding SDE is \(dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\,d\mathbf{B}_t\).

The drift must provide one rate for each state coordinate, so it is an \(n\)-vector. The diffusion must specify how each of the \(m\) noise coordinates affects each of the \(n\) state coordinates, so it is an \(n\)-by-\(m\) matrix. Its \(i\)th row describes the noise coefficients for state coordinate \(i\); its \(j\)th column describes the contribution of Brownian coordinate \(j\) across the state.

The Integral Form and Component Expansion

The differential notation is shorthand for an integral equation. In this tutorial, the stochastic integrals are Ito integrals. For each row \(i\), the process \(s\mapsto \sigma_{i\cdot}(s,X_s)\) must be progressively measurable and satisfy \(\mathbb{E}\int_0^T\|\sigma_{i\cdot}(s,X_s)\|^2\,ds<\infty\), where \(\|\cdot\|\) is the Euclidean norm. The drift must be measurable along \(X\) and satisfy \(\int_0^T\|b(s,X_s)\|\,ds<\infty\) almost surely. These conditions make the displayed integrals meaningful on \([0,T]\); they do not assert that a solution exists.

X_t=X_0+\int_0^t b(s,X_s)\,ds+\int_0^t \sigma(s,X_s)\,d\mathbf{B}_s,\qquad 0\le t\le T.

The matrix-valued stochastic integral is defined row by row. For each \(i=1,\ldots,n\), its \(i\)th coordinate is the sum of \(m\) scalar Ito integrals, each with a progressively measurable integrand whose expected squared integral over \([0,T]\) is finite. Writing the entries as \(b_i\) and \(\sigma_{ij}\) gives the equivalent coordinate equations.

dX_t^i=b_i(t,X_t)\,dt+\sum_{j=1}^{m}\sigma_{ij}(t,X_t)\,dB_t^j,\qquad i=1,\ldots,n.

Proposition: The vector differential equation with \(b\in\mathbb{R}^n\), \(\sigma\in\mathbb{R}^{n\times m}\), and \(d\mathbf{B}\in\mathbb{R}^m\) has exactly the component equations above. Conversely, those component equations assemble into the vector equation when they use the same \(n\)-dimensional state and the same \(m\)-dimensional Brownian motion.

Proof. By matrix multiplication, the \(i\)th coordinate of \(\sigma(t,X_t)d\mathbf{B}_t\) is \(\sum_{j=1}^m\sigma_{ij}(t,X_t)dB_t^j\). The \(i\)th coordinate of the drift is \(b_i(t,X_t)dt\), which gives the displayed component equation for each \(i\). Conversely, stacking those \(n\) equations places the \(n\) drift coordinates into one vector and the coefficients \(\sigma_{ij}\) into an \(n\)-by-\(m\) matrix; the resulting matrix product has precisely the given noise sums. The integral statement follows by the same row-by-row construction, with the stated integrability conditions ensuring each Ito integral is defined. \(\square\)

Scalar and Vector Shapes

When \(n=1\), the drift is scalar, but the diffusion need not be. With \(m\) Brownian coordinates, \(\sigma(t,X_t)\) is a \(1\)-by-\(m\) row, equivalently an \(m\)-component list of coefficients. When \(m=1\), the diffusion is an \(n\)-by-\(1\) column. Only when both dimensions are one is the diffusion coefficient a scalar.

Worked Example: A Two-Dimensional State with Three Noise Coordinates

Let \(X_t=(X_t^1,X_t^2)^\mathsf{T}\) take values in \(D=\mathbb{R}^2\), and let \(\mathbf{B}_t\) have three coordinates. Suppose \(b(t,x)=(t+x_1,\ x_1-x_2)^\mathsf{T}\) and \(\sigma(t,x)=\begin{pmatrix}x_1&t&1\\x_2&1&t\end{pmatrix}\). Expand the equation at \(t=1\), \(x=(2,-1)^\mathsf{T}\).

First evaluate the drift: \(b(1,(2,-1))=(1+2,\ 2-(-1))^\mathsf{T}=(3,3)^\mathsf{T}\). The diffusion matrix is \(\begin{pmatrix}2&1&1\\-1&1&1\end{pmatrix}\), with two rows because there are two state coordinates and three columns because there are three Brownian coordinates. Multiplying each row by \(d\mathbf{B}_t\) gives

\begin{aligned} dX_t^1&=3\,dt+2\,dB_t^1+dB_t^2+dB_t^3,\\ dX_t^2&=3\,dt-dB_t^1+dB_t^2+dB_t^3. \end{aligned}

Check: each equation has one drift coordinate and a sum with three Brownian increments. For a further matrix check, the row dot products are \(2^2+1^2+1^2=6\), \((-1)^2+1^2+1^2=3\), and \(2(-1)+1\cdot1+1\cdot1=0\). Thus \(\sigma\sigma^\mathsf{T}=\begin{pmatrix}6&0\\0&3\end{pmatrix}\) at this time and state.

The state set \(D\) is part of the coefficient specification. If coefficients are specified only for \(x\in D\), their values outside \(D\) are not defined by that specification. A proposed process used in the SDE must take values where the coefficients are defined. This observation is about the meaning of the equation, not a claim that a process remains in \(D\). For example, writing \(D=(0,\infty)\) does not by itself prove that a solution stays positive.

Worked Example: One State Coordinate Driven by Two Brownian Motions

Take \(D=(0,\infty)\), one state coordinate, and two Brownian coordinates. Define \(b(t,x)=-2x\) and \(\sigma(t,x)=(x,\ 2)\) for \(x\in D\). Write the equation and evaluate its coefficients at \(x=3\).

The drift maps into \(\mathbb{R}\), while the diffusion maps into \(\mathbb{R}^{1\times2}\). At \(x=3\), \(b(t,3)=-2\cdot3=-6\) and \(\sigma(t,3)=(3,2)\). Therefore the differential equation at that state has the form

dX_t=-6\,dt+3\,dB_t^1+2\,dB_t^2.

Check: the drift has one entry, matching the one-dimensional state. The diffusion has two entries, matching the two Brownian increments, and their sum is scalar, as required for \(dX_t\). These coefficient evaluations do not establish existence of a solution or that it stays in \((0,\infty)\).

Dimensional Consistency and Units

Vector dimensions and physical units are related checks, but they are not the same check. The matrix shapes ensure that sums and products have the right number of coordinates. If time has units of time and the state has units of \(X\), then \(dt\) has units of time and Brownian increments have units of the square root of time. Consequently the drift coefficient has units \(X/\text{time}\), while each diffusion entry has units \(X/\sqrt{\text{time}}\). This makes both terms on the right have units of state.

The shorthand multiplication rules \(dB_t^j\,dB_t^k=\delta_{jk}\,dt\), \(dB_t^j\,dt=0\), and \(dt\,dt=0\) also show how the diffusion shape enters second-order calculations. In particular, the coefficient-level quadratic variation rate for state coordinates \(i,k\) is the row dot product \(\sum_{j=1}^m\sigma_{ij}\sigma_{kj}\).

Proposition: Under the Ito multiplication rules, the diffusion contribution to the quadratic covariation of coordinates \(X^i\) and \(X^k\) is \(\sum_{j=1}^m\sigma_{ij}(t,X_t)\sigma_{kj}(t,X_t)\,dt\). In matrix form, the matrix of these rates is \(\sigma(t,X_t)\sigma(t,X_t)^\mathsf{T}\), an \(n\)-by-\(n\) matrix.

Proof. In coordinate \(i\), the diffusion differential is \(\sum_{j=1}^m\sigma_{ij}\,dB_t^j\). Multiplying the diffusion differentials for coordinates \(i\) and \(k\), and applying \(dB_t^j\,dB_t^\ell=\delta_{j\ell}dt\), gives

\left(\sum_{j=1}^{m}\sigma_{ij}\,dB_t^j\right) \left(\sum_{\ell=1}^{m}\sigma_{k\ell}\,dB_t^\ell\right) =\sum_{j=1}^{m}\sum_{\ell=1}^{m}\sigma_{ij}\sigma_{k\ell}\delta_{j\ell}\,dt =\sum_{j=1}^{m}\sigma_{ij}\sigma_{kj}\,dt.

Terms involving \(dt\) and a Brownian increment vanish by \(dB_t^j\,dt=0\), and terms with two \(dt\)'s vanish by \(dt\,dt=0\). The \((i,k)\) entry of \(\sigma\sigma^\mathsf{T}\) is exactly the sum obtained, proving the matrix statement. \(\square\)

Worked Example: Correcting a Mismatched Diffusion Shape

Suppose the state has three coordinates and the driving Brownian motion has two coordinates. Let \(b=(1,0,-1)^\mathsf{T}\) and \(\sigma=\begin{pmatrix}1&0\\0&2\\1&-1\end{pmatrix}\). Write the noise term and identify the error in trying to use a \(2\)-by-\(3\) diffusion matrix instead.

The correct matrix is \(3\)-by-\(2\). Its three rows give the noise contributions \(dB_t^1\), \(2dB_t^2\), and \(dB_t^1-dB_t^2\), respectively. Thus

\begin{aligned} dX_t^1&=dt+dB_t^1,\\ dX_t^2&=2\,dB_t^2,\\ dX_t^3&=-dt+dB_t^1-dB_t^2. \end{aligned}

A \(2\)-by-\(3\) matrix would have the wrong shape: multiplying it by the \(2\)-component column \(d\mathbf{B}_t\) is undefined, because the inner dimensions \(3\) and \(2\) do not match. Check: the displayed diffusion has three rows, matching the three equations, and two columns, matching \(B^1\) and \(B^2\). The drift entries \(1,0,-1\) also provide exactly one rate for each equation.

Common Pitfalls

A frequent mistake is to use the word “diffusion coefficient” as if it must always be a scalar. It is scalar only for a one-dimensional state driven by one Brownian coordinate. For a scalar state and \(m\) noise coordinates it is a row with \(m\) entries; for an \(n\)-dimensional state and one noise coordinate it is a column with \(n\) entries.

Another error is to match the diffusion matrix to the state dimension alone. The number of rows is determined by the state, but the number of columns is determined by the noise. Finally, correct dimensions do not prove existence, uniqueness, or state-space preservation. Those require additional hypotheses and arguments; a dimension check verifies that the proposed equation is structurally meaningful, not that it has a solution.

Check Your Understanding

For each question, identify the state dimension and noise dimension before checking the coefficient shapes.

If \(X_t\in\mathbb{R}^4\) is driven by a two-dimensional Brownian motion, what are the dimensions of its drift and diffusion coefficients?
Write a scalar SDE driven by three Brownian coordinates in component notation using drift \(b\) and diffusion row \((\sigma_1,\sigma_2,\sigma_3)\).
For an \(n\)-dimensional state driven by one Brownian motion, what shape must the diffusion coefficient have?
Explain why a diffusion coefficient with three rows and two columns cannot multiply a three-coordinate Brownian increment column.
If a state has units \(X\), what units must the drift and diffusion coefficients have so that both SDE terms have units \(X\)?