[rk-mlu]

Third test problem of milestone M1 (roadmap: "Brownian motion on the circle as third test problem"). Independent of #16; based on main.

Problem

$$ \mathrm{d}X(t) = -\tfrac12 \sigma^2 X(t),\mathrm{d}t + \sigma J X(t),\mathrm{d}W(t), \qquad J = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix}, $$

with $d = 2$, $m = 1$ (scalar, multiplicative noise). The exact solution is $X(t) = R(\sigma(W(t) - W(t_0))),x_0$, a rotation of $x_0$, so $|X(t)| = |x_0|$. Defaults: $\sigma = 1$, $x_0 = (1, 0)$, $t_0 = 0$. Registry name brownian_motion_on_circle.

Euler–Maruyama leaves the circle: on every path $|X_h^{n+1}|^2 = |X_h^n|^2\big((1 - \sigma^2 h_n/2)^2 + \sigma^2 \Delta W_n^2\big)$, and on a uniform grid $\mathbb{E}|X_h^N|^2 = |x_0|^2 (1 + \sigma^4 h^2/4)^N$. Both identities were checked numerically before they went onto the theory page, and both are tested.

There is no legacy script for this problem in the archive; the scripts unitcirc*.py sample the uniform distribution on the circle.

Tests

Kind	What
Unit	coefficients and rotation by hand; equality with the commuting matrix formula $\exp((A - B^2/2)t + BW),x_0$ (truncated series); broadcasting; float32 initial value; parameter validation
Property	the per-step radius identity of Euler–Maruyama for random grids, $\sigma$ and starting angles
PyTorch	solution and exact solution equal NumPy
Statistical (slow)	strong order 1/2 (with GBM and the linear system); $\mathbb{E}|X_h^N|^2$ against the closed form within five standard errors, with the drift resolved by more than ten

Coverage 100 % with and without PyTorch and matplotlib; Ruff, mypy, codespell and Sphinx -W pass.

Please check

• A literature reference for the problem (the theory page says that one is still to be added).

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