Stochastic Differential Equations: A Crash Course - The Foundation Inference Model Program

This section provides a theoretical foundation for understanding Stochastic Differential Equations (SDEs) and introduces the concepts needed for the practical tutorial. This content is adapted from Appendix B of the corresponding publication Seifner et al. (2025).

Mathematical Foundation¶

A $d$ -dimensional stochastic process $x(t)$ follows an Itô stochastic differential equation (SDE) if it satisfies:

x_i(\overline{t})=x_i(\underline{t})+\int_{\underline{t}}^{\overline{t}} f_i(x(t'),t')dt'+\sum_{j}^{m}\int_{\underline{t}}^{\overline{t}} G_{ij}(x(t'),t')dW_j(t')

(1)

for all $i\leq d,\ \underline{t}\leq \overline{t}$ and some vector-valued drift function $f:\mathbb{R}^d\times \mathbb{R}^+\to\mathbb{R}^d$ and diffusion matrix $G:\mathbb{R}^d\times\mathbb{R}^+\to\mathbb{R}^{d\times m}$ , where $W:\mathbb{R}^+\to\mathbb{R}^m$ is a standard $m$ -dimensional Wiener process.

In differential notation, this is commonly written as:

dx(t)=f(x(t),t)dt+G(x(t),t)dW(t)

(2)

Model Capabilities and Assumptions¶

Our Foundation Inference Model (FIM) for SDEs can estimate both the drift function $f$ and diffusion function $G$ in a zero-shot manner directly from observed trajectory data. The model assumes diagonal diffusion, i.e.

G(x)=\text{diag}(\sqrt{g_1(x)},\dots,\sqrt{g_d(x)})

(3)

and therefore returns the vector field $(\sqrt{\hat{g}_1(x)},\dots,\sqrt{\hat{g}_d(x)})$ .

Furthermore this model assumes purely state-dependent drift and diffusion!

References¶

Seifner, P., Cvejoski, K., Berghaus, D., Ojeda, C., & Sánchez, R. J. (2025). In-Context Learning of Stochastic Differential Equations with Foundation Inference Models. The Thirty-Ninth Annual Conference on Neural Information Processing Systems. https://openreview.net/forum?id=ceCJPoZOKJ