Approximations of the solutions of a stochastic differential equation using Dirichlet process mixtures and Gaussian mixtures

Abstract

Stochastic differential equations arise in a variety of contexts. There are many techniques for approximation of the solutions of these equations that include statistical methods, numerical methods, and other approximations. This article implements a parametric and a nonparametric method to approximate the probability density of the solutions of stochastic differential equation from observations of a discrete dynamic system. To achieve these objectives, mixtures of Dirichlet process and Gaussian mixtures are considered. The methodology uses computational techniques based on Gaussian mixtures filters, nonparametric particle filters and Gaussian particle filters to establish the relationship between the theoretical processes of unobserved and observed states. The approximations obtained by this proposal are attractive because the hierarchical structures used for modeling are flexible, easy to interpret and computationally feasible. The methodology is illustrated by means of two problems: the synthetic Lorenz model and a real model of rainfall. Experiments show that the proposed filters provide satisfactory results, demonstrating that the algorithms work well in the context of processes with nonlinear dynamics which require the joint estimation of states and parameters. The estimated measure of goodness of fit validates the estimation quality of the algorithms.