A Milstein-type method for highly non-linear non-autonomous time-changed stochastic differential equations

TL;DR

This paper introduces a Milstein-type numerical method tailored for highly non-linear, non-autonomous time-changed SDEs, analyzing its strong convergence and convergence order under specific growth and continuity conditions.

Contribution

It develops a novel Milstein-type scheme for complex time-changed SDEs with super-linear growth and H"older continuity, providing convergence analysis and order.

Findings

The method achieves strong convergence in finite time.

Convergence order is explicitly derived.

Applicable to highly non-linear, non-autonomous SDEs.

Abstract

A Milstein-type method is proposed for some highly non-linear non-autonomous time-changed stochastic differential equations (SDEs). The spatial variables in the coefficients of the time-changed SDEs satisfy the super-linear growth condition and the temporal variables obey some H\"older's continuity condition. The strong convergence in the finite time is studied and the convergence order is obtained.