Behavioral Equations KirmansAnts

Step Equations

Run substeps Euler-Maruyama micro-steps in Lamperti space. Uses the Lamperti transform \(\phi = \arcsin(2 x - 1)\) from Moran, Fosset, Benzaquen, Bouchaud (2020). The diffusion in \(\phi\) is constant \(= \sqrt{2\mu}\), so Euler-Maruyama in \(\phi\) is Lipschitz and recovers the standard \(O(dt)\) weak rate. The \(\phi\)-domain is \([-\pi/2,\,+\pi/2]\); reflective BC in \(\phi\)-space is the geometrically correct boundary condition. \(x\) is clipped to \([\epsilon, 1 - \epsilon]\) with \(\epsilon = 10^{-7}\) before the forward map and after the inverse map. The clip sits above the float32 precision of the side-buffer near the boundary and far below any KDE bandwidth used downstream, so the clip is invisible to consumers. _micro_trajectory stores \(x\) (not \(\phi\)); the inverse map is applied at every substep so downstream consumers stay method-agnostic.

\begin{align} \phi &= \arcsin(2 x - 1), \\ d\phi_t &= \mu_\Phi(\phi_t)\, dt + \sigma_\Phi\, dW_t, \quad \sigma_\Phi = \sqrt{2\mu}, \\ \mu_\Phi(\phi) &= -(2\rho - \mu)\tan(\phi) = -\,\frac{(2\rho - \mu)\,(2 x - 1)}{2\sqrt{x(1-x)}}, \\ x &= \tfrac{1}{2}\bigl(1 + \sin\phi\bigr). \end{align}

1. Step

Run substeps Euler-Maruyama micro-steps in Lamperti space. Uses the Lamperti transform \(\phi = \arcsin(2 x - 1)\) from Moran, Fosset, Benzaquen, Bouchaud (2020). The diffusion in \(\phi\) is constant \(= \sqrt{2\mu}\), so Euler-Maruyama in \(\phi\) is Lipschitz and recovers the standard \(O(dt)\) weak rate. The \(\phi\)-domain is \([-\pi/2,\,+\pi/2]\); reflective BC in \(\phi\)-space is the geometrically correct boundary condition. \(x\) is clipped to \([\epsilon, 1 - \epsilon]\) with \(\epsilon = 10^{-7}\) before the forward map and after the inverse map. The clip sits above the float32 precision of the side-buffer near the boundary and far below any KDE bandwidth used downstream, so the clip is invisible to consumers. _micro_trajectory stores \(x\) (not \(\phi\)); the inverse map is applied at every substep so downstream consumers stay method-agnostic.

\begin{align} \phi &= \arcsin(2 x - 1), \\ d\phi_t &= \mu_\Phi(\phi_t)\, dt + \sigma_\Phi\, dW_t, \quad \sigma_\Phi = \sqrt{2\mu}, \\ \mu_\Phi(\phi) &= -(2\rho - \mu)\tan(\phi) = -\,\frac{(2\rho - \mu)\,(2 x - 1)}{2\sqrt{x(1-x)}}, \\ x &= \tfrac{1}{2}\bigl(1 + \sin\phi\bigr). \end{align}