src/DiffEq/Dirichlet/GeneralizedDirichlet.hpp file

Lochner's generalized Dirichlet SDE.


In a nutshell, the equation integrated governs a set of scalars, $0 \le Y_i$ , $i=1,\dots,K$ , $\sum_{i=1}^KY_i\le1$ , as

\[ \begin{split} \mathrm{d}Y_i(t) = \frac{\mathcal{U}_i}{2}\left\{ b_i\Big[S_i \mathcal{Y}_K - (1-S_i)Y_i\Big] + Y_i\mathcal{Y}_K \sum_{j=i}^{K-1}\frac{c_{ij}}{\mathcal{Y}_j}\right\}\mathrm{d}t + \sqrt{\kappa_i Y_i \mathcal{Y}_K \mathcal{U}_i}\mathrm{d}W_i(t), \\ \qquad i=1,\dots,K, \end{split} \]
\[ \begin{split} \mathrm{d}Y_i(t) = \frac{\mathcal{U}_i}{2}\left\{ b_i\Big[S_i \mathcal{Y}_K - (1-S_i)Y_i\Big] + Y_i\mathcal{Y}_K \sum_{j=i}^{K-1}\frac{c_{ij}}{\mathcal{Y}_j}\right\}\mathrm{d}t \\ + \sqrt{\kappa_i Y_i \mathcal{Y}_K \mathcal{U}_i}\mathrm{d}W_i(t), \qquad i=1,\dots,K, \end{split} \]

where $\mathrm{d}W_i(t)$ is an isotropic vector-valued Wiener process with independent increments. The statistically stationary solution of the above coupled system of nonlinear stochastic differential equations is the generalized Dirichlet distribution,

\[ \newcommand{\bv}[1]{{\mbox{$\mathbf{#1}$}}} G(\bv{Y},\bv{\alpha},\bv{\beta}) = \prod_{i=1}^K\frac{\Gamma(\alpha_i+\beta_i)}{\Gamma(\alpha_i) \Gamma(\beta_i)}Y_i^{\alpha_i-1} \mathcal{Y}_i^{\gamma_i} \qquad \mathrm{with} \qquad \mathcal{Y}_i = 1-\sum_{k=1}^i Y_k, \]
\[ \begin{split} \newcommand{\bv}[1]{{\mbox{$\mathbf{#1}$}}} G(\bv{Y},\bv{\alpha},\bv{\beta}) = \prod_{i=1}^K\frac{\Gamma(\alpha_i+\beta_i)}{\Gamma(\alpha_i) \Gamma(\beta_i)}Y_i^{\alpha_i-1} \mathcal{Y}_i^{\gamma_i} \\ \mathrm{with} \qquad \mathcal{Y}_i = 1-\sum_{k=1}^i Y_k, \end{split} \]

provided the coefficients, $b_i\!>\!0$ , $\kappa_i\!>\!0$ , $0\!<\!S_i\!<\!1$ , and $c_{ij}$ , with $c_{ij}\!=\!0$ for $i\!>\!j$ , $i,j\!=\!1,\dots,K\!-\!1$ , satisfy

\[ \begin{split} \alpha_i & = \frac{b_i}{\kappa_i}S_i, \qquad i=1,\dots,K,\\ 1-\gamma_i & = \frac{c_{1i}}{\kappa_1} = \dots = \frac{c_{ii}}{\kappa_i}, \qquad i=1,\dots,K-1,\\ 1+\gamma_K & = \frac{b_1}{\kappa_1}(1-S_1) = \dots = \frac{b_K}{\kappa_K}(1-S_K). \end{split} \]

Here $\mathcal{U}_i = \prod_{j=1}^{K-i}\mathcal{Y}_{K-j}^{-1}$ , $\alpha_i>0$ , and $\beta_i>0$ are parameters, while $\gamma_i=\beta_i-\alpha_{i+1}-\beta_{i+1}$ for $i=1,\dots,K-1$ , and $\gamma_K=\beta_K-1$ . $\Gamma(\cdot)$ denotes the gamma function. To keep the invariant distribution generalized Dirichlet, the above set of constraints on the coefficients must be satisfied. For more details on the generalized Dirichlet SDE, see


namespace walker
Walker declarations and definitions.


template<class Init, class Coefficients>
class walker::GeneralizedDirichlet
Lochner's generalized Dirichlet SDE used polymorphically with DiffEq.