Convergence of numerical methods for stochastic differential equations in mathematical finance

TL;DR

This paper reviews recent advances in the convergence analysis of numerical methods for stochastic differential equations in financial models, especially those with non-globally Lipschitz coefficients like Heston and Cox-Ingersoll-Ross models.

Contribution

It provides a comprehensive review of new convergence results for numerical schemes applied to complex financial SDEs with unbounded coefficients.

Findings

New convergence results for non-globally Lipschitz SDEs

Analysis of strong and weak convergence in financial models

Discussion of positivity-preserving numerical methods

Abstract

Many stochastic differential equations that occur in financial modelling do not satisfy the standard assumptions made in convergence proofs of numerical schemes that are given in textbooks, i.e., their coefficients and the corresponding derivatives appearing in the proofs are not uniformly bounded and hence, in particular, not globally Lipschitz. Specific examples are the Heston and Cox-Ingersoll-Ross models with square root coefficients and the Ait-Sahalia model with rational coefficient functions. Simple examples show that, for example, the Euler-Maruyama scheme may not converge either in the strong or weak sense when the standard assumptions do not hold. Nevertheless, new convergence results have been obtained recently for many such models in financial mathematics. These are reviewed here. Although weak convergence is of traditional importance in financial mathematics with its emphasis on expectations of functionals of the solutions, strong convergence plays a crucial role in Multi Level Monte Carlo methods, so it and also pathwise convergence will be considered along with methods which preserve the positivity of the solutions.