Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integrators

TL;DR

This paper investigates the strong convergence rates of approximation schemes for a class of SDEs involving random time changes and non-standard Lipschitz conditions, providing new analytical techniques.

Contribution

It introduces novel methods to analyze strong convergence for SDEs with random time changes and time-varying Lipschitz bounds, diverging from traditional duality-based approaches.

Findings

Established convergence rates for approximation schemes.

Developed criteria for exponential moments of random time changes.

Provided analytical tools for SDEs with complex drift structures.

Abstract

The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time change given by an inverse subordinator. SDEs to be considered are unique in two different aspects: i) they contain two drift terms, one driven by the random time change and the other driven by a regular, non-random time variable; ii) the standard Lipschitz assumption is replaced by that with a time-varying Lipschitz bound. The difficulty imposed by the first aspect is overcome via an approach that is significantly different from a well-known method based on the so-called duality principle. On the other hand, the second aspect requires the establishment of a criterion for the existence of exponential moments of functions of the random time change.