Improving Monte Carlo simulations by Dirichlet forms

TL;DR

This paper introduces a method to enhance Monte Carlo simulations by utilizing Dirichlet forms, enabling more efficient estimation of expectations and densities when additional structure is available.

Contribution

It provides a novel approach to improve Monte Carlo methods using Dirichlet forms and explicit formulas for density estimation on various probability spaces.

Findings

Enhanced Monte Carlo simulations with Dirichlet forms improve density estimation.

Explicit formulas enable faster convergence to true densities.

Applicable to Wiener, Poisson, and Monte Carlo spaces.

Abstract

Equipping the probability space with a local Dirichlet form with square field operator \Gamma and generator A allows to improve Monte Carlo simulations of expectations and densities as soon as we are able to simulate a random variable X together with \Gamma[X] and A[X]. We give examples on the Wiener space, on the Poisson space and on the Monte Carlo space. When X is real-valued we give an explicit formula yielding the density at the speed of the law of large numbers.