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

Abstract: This paper investigates the convergence rates of two Euler-type methods for a class of time-changed stochastic differential equations with super-linearly growing drift and diffusion coefficients. Building upon existing research, we adapt the backward Euler method to time-changed stochastic differential equations where both coefficients exhibit super-linear growth and introduce an explicit counterpart, the projected Euler method. It is shown that both methods achieve the optimal strong convergence rate of order 1/2 in the mean-square sense for this class of equations. Numerical simulations confirm the theoretical findings