src/DiffEq/Beta.h file

System of beta SDEs.

Contents

This file implements the time integration of a system of stochastic differential equations (SDEs) with linear drift and quadratic diagonal diffusion, whose invariant is the joint beta distribution.

In a nutshell, the equation integrated governs a set of scalars, $0\!\le\!Y_\alpha$ , $\alpha\!=\!1,\dots,N$ , as

\[ \mathrm{d}Y_\alpha(t) = \frac{b_\alpha}{2}\left(S_\alpha - Y_\alpha\right) \mathrm{d}t + \sqrt{\kappa_\alpha Y_\alpha(1-Y_\alpha)} \mathrm{d}W_\alpha(t), \qquad \alpha=1,\dots,N \]
\[ \begin{split} \mathrm{d}Y_\alpha(t) = \frac{b_\alpha}{2}\left(S_\alpha - Y_\alpha\right) \mathrm{d}t + \sqrt{\kappa_\alpha Y_\alpha(1-Y_\alpha)} \mathrm{d}W_\alpha(t), \\ \alpha=1,\dots,N \end{split} \]

with parameter vectors $b_\alpha > 0$ , $\kappa_\alpha > 0$ , and $0 < S_\alpha < 1$ . Here $\mathrm{d}W_\alpha(t)$ is an isotropic vector-valued Wiener process with independent increments. The invariant distribution is the joint beta distribution. This system of SDEs consists of N independent equations. For more on the beta SDE, see https://doi.org/10.1080/14685248.2010.510843.

Namespaces

namespace walker
Walker declarations and definitions.

Classes

template<class Init, class Coefficients>
class walker::Beta
Beta SDE used polymorphically with DiffEq.