Stochastic Differential Equations: A Crash Course

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 [SCB+25].

Mathematical Foundation

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

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

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:

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

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.

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

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!