Simulation of diffusions by means of importance sampling paradigm

TL;DR

This paper introduces a novel Monte Carlo simulation method for stochastic differential equations that combines importance sampling with random walk techniques, improving accuracy near complex boundaries and aiding in rare event simulation.

Contribution

It presents a new importance sampling-based Monte Carlo approach that enhances diffusion simulation accuracy and efficiency, especially with complex boundary conditions.

Findings

More accurate diffusion approximation than Euler scheme

Easier computation of weights from Brownian motion density

Effective for variance reduction and rare event simulation

Abstract

The aim of this paper is to introduce a new Monte Carlo method based on importance sampling techniques for the simulation of stochastic differential equations. The main idea is to combine random walk on squares or rectangles methods with importance sampling techniques. The first interest of this approach is that the weights can be easily computed from the density of the one-dimensional Brownian motion. Compared to the Euler scheme this method allows one to obtain a more accurate approximation of diffusions when one has to consider complex boundary conditions. The method provides also an interesting alternative to performing variance reduction techniques and simulating rare events.