Strong and weak convergence rates of logarithmic transformed truncated EM methods for SDEs with positive solutions

Abstract

To inherit numerically the positivity of stochastic differential equations (SDEs) with non-globally Lipschitz coefficients, we devise a novel explicit method, called logarithmic transformed truncated Euler–Maruyama method. There is however a price to be paid for the preserving positivity, namely that the logarithmic transformation would cause the coefficients of the transformed SDEs growing super-linearly or even exponentially, which makes the strong and weak convergence analysis more complicated. Based on the exponential integrability, truncation techniques and some other arguments, we show that the strong convergence rate of the underlying numerical method is 1/2, and the weak convergence rate can be arbitrarily close to 1. To the best of our knowledge, this is the first result establishing the weak convergence rate of numerical methods for the general SDEs with positive solutions. Numerical experiments are finally reported to confirm our theoretical results.