Weak convergence order of stochastic theta method for SDEs driven by time-changed L\'{e}vy noise

TL;DR

This paper analyzes the weak convergence order of the stochastic theta method for SDEs driven by time-changed Lévy noise, addressing complexities introduced by the random time change and implicit drift correction.

Contribution

It extends weak convergence analysis from Euler–Maruyama to the stochastic theta method for time-changed Lévy SDEs, incorporating inverse subordinator approximation and duality principles.

Findings

Established weak convergence order one for non-time-changed Lévy SDEs.

Derived the weak convergence order for time-changed Lévy SDEs with θ in [0,1].

Numerical experiments support the theoretical convergence results.

Abstract

This paper studies the weak convergence order of the stochastic theta method for stochastic differential equations (SDEs) driven by time-changed L\'{e}vy noise under global Lipschitz and linear growth conditions. In contrast to classical L\'{e}vy-driven SDEs, the presence of a random time change makes the weak error analysis involve both the discretization error of the underlying equation and the approximation error of the random clock. Moreover, compared with explicit Euler--Maruyama method, the implicit drift correction in the stochastic theta method makes the associated weak error analysis substantially more delicate. To address these difficulties, we first establish a global weak convergence estimate of order one for the stochastic theta method applied to the corresponding non-time-changed L\'{e}vy SDEs on the infinite time interval by means of the Kolmogorov backward partial integro differential equations. Incorporating the approximation of the inverse subordinator together with the duality principle, we derive the weak convergence order of the stochastic theta method with $\theta \in [0,1]$ in the time-changed L\'{e}vy setting. The result advances the currently available weak convergence analysis beyond the Euler--Maruyama method to the more general class of stochastic theta method, and establishes a workable route from the weak analysis of the underlying non-time-changed L\'{e}vy equation to the corresponding time-changed problem. Finally, numerical experiments are presented to further support the theoretical findings.