Convergence and Stability of an Explicit Method for Autonomous Time-Changed Stochastic Differential Equations with Super-Linear Coefficients

Abstract

In this paper, numerical methods for the time-changed stochastic differential equations of the form $$d\Upsilon(t)=a(\Upsilon(t))dt+b(\Upsilon(t))dE(t)+\sigma(\Upsilon(t))dB(E(t))$$ are investigated, where all the coefficients $a(·),$ $b(·)$ and $\sigma(·)$ are allowed to contain some super-linearly growing terms. An explicit method is proposed by using the idea of truncating terms that grow too fast. Strong convergence in the finite time of the proposed method is proved and the convergence rate is obtained. The proposed method is also proved to be able to reproduce the asymptotic stability of the underlying equation in the almost sure sense. Simulations are provided to demonstrate the theoretical results.