Formulating SDEs from Model Statements

From Model Language to an SDE

A stochastic differential equation is a compact way to record how a system changes: one term describes its accumulated drift, and another describes its accumulated random forcing. Formulating the equation requires more than replacing the words “rate” and “noise” with symbols. One must specify the state dimension, the noise dimension, the coefficients’ dependence on time and state, and the initial condition.

Throughout, work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion relative to the filtration. For a scalar equation, write the Brownian motion as \(B_t\). For a system with \(m\) noise sources, write \(\mathbf{B}_t=(B_t^1,\ldots,B_t^m)^\mathsf{T}\) for an \(m\)-dimensional Brownian motion whose components are independent standard Brownian motions. All stochastic integrals below are Ito integrals. On a finite interval \([0,T]\), each integrand \(H\) is progressively measurable and belongs to \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\); for vector integrands, this condition is \(\mathbb{E}\int_0^T\|H_s\|^2\,ds<\infty\).

For a \(d\)-dimensional state and \(m\) noise sources, the corresponding notation is \(d\mathbf{X}_t=\mathbf{b}(t,\mathbf{X}_t)\,dt+G(t,\mathbf{X}_t)\,d\mathbf{B}_t\). Here \(\mathbf{b}\) has \(d\) components and \(G\) is a \(d\)-by-\(m\) matrix. Its integral form is \(\mathbf{X}_t=\boldsymbol{\xi}+\int_0^t\mathbf{b}(s,\mathbf{X}_s)\,ds+\int_0^tG(s,\mathbf{X}_s)\,d\mathbf{B}_s\). In the last integral, each row of \(G\) is an Ito integrand; each entry must be progressively measurable and square-integrable on the stated horizon.

A Translation Checklist

A coefficient multiplying \(dB_t\) is not an ordinary rate: Brownian increments have variance equal to the time increment, so diffusion coefficients describe the scale of random fluctuations. If verbal instructions specify correlated noise, first identify the Brownian drivers and their covariance structure, then encode that structure in the diffusion matrix. Do not add a separate noise term merely because the same source has been written in two different ways.

Proof. Starting from the proposed differential notation, the definition of an SDE on \([0,T]\) gives exactly the displayed integral equation. The drift integral exists by the assumed integrability of \(b\); the Ito integral exists because \(g\) is progressively measurable and square-integrable. Conversely, given the integral equation, the increment between times \(u\) and \(t\), for \(0\le u\le t\le T\), is \(\int_u^t b_s\,ds+\int_u^t g_s\,dB_s\). This is precisely what the differential notation \(dX_t=b_t\,dt+g_t\,dB_t\) denotes. At \(t=0\), both integrals vanish, so \(X_0=\xi\). \(\square\)

Worked Formulations

Proof. Each increment \(B_t^j-B_s^j\) has mean zero and variance \(t-s\). For distinct \(j,k\), independence gives \(\mathbb{E}[(B_t^j-B_s^j)(B_t^k-B_s^k)]=0\). Expanding the square of the sum therefore gives \(\sum_j c_j^2(t-s)\) from the diagonal terms and zero from every cross term. This is the stated variance. If all coefficients are zero, both the sum and its variance are zero, so the formula covers that case as well. \(\square\)

Consistency Checks and a Common Mistake

Before accepting a formulation, check that the drift has the same dimension as the state, the diffusion matrix has one column per Brownian driver, and its rows match the state dimension. Check that the drift is integrated against \(dt\), the random forcing against \(dB_t\), and that the integral equation returns the stated initial condition at \(t=0\). Also check that each Ito integrand is progressively measurable and meets the required square-integrability condition on the horizon under discussion. Such checks establish that the equation is well-formed; by themselves they do not establish that a solution exists or is unique.

Common mistake—putting a random forcing rate against time. If a model states \(dX_t=2\,dt+3\,dB_t\), a wrong transcription is \(dX_t=(2+3)\,dt\). This treats the Brownian forcing as a deterministic rate. The correction is to retain the distinct integrators: \(X_t=X_0+\int_0^t2\,ds+\int_0^t3\,dB_s\). For a constant diffusion coefficient \(3\), the second integral has variance \(9t\), while a deterministic contribution \(3t\) has variance zero. Thus merging the terms would change the model, not just its notation.

Another dimensional trap is to write a two-dimensional noise coefficient as a single scalar without saying how it acts on the state. Specify the diffusion matrix, including its zero entries when a noise source does not affect a component. Finally, when the model gives correlated forcing, preserve its covariance: using independent Brownian motions directly for correlated sources generally changes the variance unless the correlation has been incorporated into the coefficients.