Positivity-preserving truncated Euler and Milstein methods for financial SDEs with super-linear coefficients

TL;DR

This paper introduces positivity-preserving truncated Euler and Milstein schemes for SDEs with super-linear coefficients, achieving optimal convergence rates and demonstrating effectiveness through numerical experiments.

Contribution

It develops new truncated schemes that preserve positivity for SDEs with super-linear coefficients and establishes their strong convergence rates.

Findings

Optimal strong convergence rates derived for the schemes.

Schemes effectively handle super-linear and sub-linear coefficient models.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we propose two variants of the positivity-preserving schemes, namely the truncated Euler-Maruyama (EM) method and the truncated Milstein scheme, applied to stochastic differential equations (SDEs) with positive solutions and super-linear coefficients. Under some regularity and integrability assumptions we derive the optimal strong convergence rates of the two schemes. Moreover, we demonstrate flexibility of our approaches by applying the truncated methods to approximate SDEs with super-linear coefficients (3/2 and Ai{\i}t-Sahalia models) directly and also with sub-linear coefficients (CIR model) indirectly. Numerical experiments are provided to verify the effectiveness of the theoretical results.