Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients

TL;DR

This paper introduces an explicit numerical method for SDEs with superlinear growth and one-sided Lipschitz coefficients, demonstrating strong convergence and computational efficiency over implicit schemes.

Contribution

The paper proposes a new explicit scheme that converges strongly for challenging SDEs, offering a practical and faster alternative to implicit methods.

Findings

The explicit scheme converges strongly with order one-half.

Simulations show the scheme is faster than implicit Euler.

The method handles superlinear growth coefficients effectively.

Abstract

On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.