$\alpha$-scaled strong convergence of stochastic theta method for stochastic differential equations driven by time-changed L\'evy noise beyond Lipschitz continuity

TL;DR

This paper introduces an $ ext{alpha}$-parametrized framework for analyzing the strong convergence of the stochastic theta method applied to stochastic differential equations driven by time-changed L\'evy noise, especially under local Lipschitz conditions.

Contribution

It extends existing convergence analysis to time-changed L\'evy-driven SDEs with non-global Lipschitz coefficients using an $ ext{alpha}$-parametrized approach.

Findings

Convergence order is $\min\{\eta_{F},\eta_{G},\eta_{H},\alpha/2\}$.

Explicit moment bounds for solutions are derived.

The framework applies to models in finance and biology.

Abstract

This paper develops an $\alpha$-parametrized framework for analyzing the strong convergence of the stochastic theta (ST) method for stochastic differential equations driven by time-changed L\'evy noise (TCSDEwLNs) with time-space-dependent coefficients satisfying local Lipschitz conditions. Properties of the inverse subordinator are investigated and explicit moment bounds for the exact solution are derived with jump rate incorporated. The analysis demonstrates that the ST method converges strongly with order of $min\{\eta_{F},\eta_{G},\eta_{H},\alpha/2\}$, establishing a precise relationship between numerical accuracy and the time-change mechanism. This theoretical advancement extends existing results and would facilitate applications in finance and biology where time-changed L\'evy models are prevalent.