Unbiased Monte Carlo Simulation of Diffusion Processes

TL;DR

This paper introduces an unbiased Monte Carlo simulation method for diffusion processes using an auxiliary Poisson process, ensuring convergence and finite variance even in complex multidimensional cases.

Contribution

The authors develop a novel Monte Carlo scheme that eliminates bias in diffusion process simulations while maintaining finite variance and applicability to multidimensional, nonconstant parameter models.

Findings

The method guarantees unbiased estimates of diffusion processes.

Simulation variance remains finite across all scenarios.

The approach is applicable to complex models with stochastic interest rates.

Abstract

Monte Carlo simulations of diffusion processes often introduce bias in the final result, due to time discretization. Using an auxiliary Poisson process, it is possible to run simulations which are unbiased. In this article, we propose such a Monte Carlo scheme which converges to the exact value. We manage to keep the simulation variance finite in all cases, so that the strong law of large numbers guarantees the convergence. Moreover, the simulation noise is a decreasing function of the Poisson process intensity. Our method handles multidimensional processes with nonconstant drifts and nonconstant variance-covariance matrices. It also encompasses stochastic interest rates.