1/2 order convergence rate of Euler-type methods for time-changed stochastic differential equations with super-linearly growing drift and diffusion coefficients

TL;DR

This paper demonstrates that two Euler-type numerical methods for time-changed stochastic differential equations with super-linear coefficients converge strongly at a rate of 1/2, supported by theoretical proofs and numerical simulations.

Contribution

The paper introduces a backward Euler method and a projected Euler method for TCSDEs, proving their strong convergence at the optimal rate of 1/2, extending their applicability.

Findings

Both methods converge strongly at rate 1/2.

Numerical simulations confirm theoretical convergence.

PEM is computationally more efficient than BEM.

Abstract

This paper investigates the strong convergence properties of two Euler-type methods for a class of time-changed stochastic differential equations (TCSDEs) with super-linearly growing drift and diffusion coefficients. Building upon existing research, we propose a backward Euler method (BEM) and introduce its explicit counterpart -- the projected Euler method (PEM). We prove that both methods converge strongly in the $L_2$-sense at the optimal rate of 1/2. This result extends the applicability of both the BEM and the PEM to a broader class of TCSDEs. Moreover, the two methods offer complementary strengths: while BEM possesses wide applicability, PEM is computationally more efficient. Numerical simulations confirm our theoretical findings and illustrate practical performance of both schemes.