An implementation of Milstein's method for general bounded diffusions

TL;DR

This paper presents a detailed implementation of Milstein's method for bounded diffusions, achieving high accuracy with manageable computational cost, and introduces novel theoretical insights linking numerical methods and Eikonal equations.

Contribution

It provides a practical, efficient implementation of Milstein's method for bounded diffusions, with theoretical innovations including a new rank-one update formula and a connection to Eikonal equations.

Findings

Method achieves (nearly) linear weak convergence rate.

Implementation is comparable in cost to popular schemes.

Numerical examples confirm accuracy and robustness.

Abstract

Despite its generality and powerful convergence properties, Milstein's method for functionals of spatially bounded stochastic differential equations is widely regarded as difficult to implement. This has likely prevented it from being utilised in applications. In this paper, we design and analyse in detail one such implementation. The presented method turns out to be on par with other, popular schemes in terms of computational cost---but with a (nearly) linear weak convergence rate under the usual smoothness requirements on coefficients and boundary. Two byproducts of theoretical interest are a new, non-standard rank-one update formula, and a connection between numerics of bounded diffusions and Eikonal equations. Three examples are worked out, confirming the accuracy and robustness of the method.