Discretizing Distributions with Exact Moments: Error Estimate and Convergence Analysis

TL;DR

This paper introduces a method for discretizing continuous distributions by minimizing relative entropy under moment constraints, providing error bounds and convergence analysis, with applications in numerical problems like portfolio optimization.

Contribution

It presents a novel approach combining maximum entropy and quadrature-based discretization, with theoretical error estimates and convergence guarantees.

Findings

Error bound matches the order of the initial quadrature formula

Discrete distribution weakly converges to the continuous distribution

Numerical examples demonstrate the method's effectiveness

Abstract

The maximum entropy principle is a powerful tool for solving underdetermined inverse problems. This paper considers the problem of discretizing a continuous distribution, which arises in various applied fields. We obtain the approximating distribution by minimizing the Kullback-Leibler information (relative entropy) of the unknown discrete distribution relative to an initial discretization based on a quadrature formula subject to some moment constraints. We study the theoretical error bound and the convergence of this approximation method as the number of discrete points increases. We prove that (i) the theoretical error bound of the approximate expectation of any bounded continuous function has at most the same order as the quadrature formula we start with, and (ii) the approximate discrete distribution weakly converges to the given continuous distribution. Moreover, we present some numerical examples that show the advantage of the method and apply to numerically solving an optimal portfolio problem.