# Strong approximation of stochastic differential equations driven by a time-changed Brownian motion with time-space-dependent coefficients

## TL;DR

This paper studies the strong convergence rate of an approximation scheme for stochastic differential equations driven by time-changed Brownian motion with coefficients depending on real time, using a novel inequality and exponential moment criteria.

## Contribution

It introduces a new analysis method for SDEs with time-dependent coefficients driven by complex time changes, extending previous work to more general settings.

## Key findings

- Established a strong convergence rate for the approximation scheme.
- Developed a Gronwall-type inequality involving stochastic drivers.
- Provided criteria for the existence of exponential moments of the time change.

## Abstract

The rate of strong convergence is investigated for an approximation scheme for a class of stochastic differential equations driven by a time-changed Brownian motion, where the random time changes $(E_t)_{t\ge 0}$ considered include the inverses of stable and tempered stable subordinators as well as their mixtures. Unlike those in the work of Jum and Kobayashi (2016), the coefficients of the stochastic differential equations discussed in this paper depend on the regular time variable $t$ rather than the time change $E_t$. This alteration makes it difficult to apply the method used in that paper. To overcome this difficulty, we utilize a Gronwall-type inequality involving a stochastic driver to control the moment of the error process. Moreover, in order to guarantee that an ultimately derived error bound is finite, we establish a useful criterion for the existence of exponential moments of powers of the random time change.