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Weak convergence order of stochastic theta method for SDEs driven by time-changed Lévy noise

Ziheng Chen, Jiao Liu, Meng Cai

Abstract

This paper studies the weak convergence order of the stochastic theta method for stochastic differential equations (SDEs) driven by time-changed Lévy noise under global Lipschitz and linear growth conditions. In contrast to classical Lévy-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 Lévy 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 $θ\in [0,1]$ in the time-changed Lévy 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 Lévy equation to the corresponding time-changed problem. Finally, numerical experiments are presented to further support the theoretical findings.

Weak convergence order of stochastic theta method for SDEs driven by time-changed Lévy noise

Abstract

This paper studies the weak convergence order of the stochastic theta method for stochastic differential equations (SDEs) driven by time-changed Lévy noise under global Lipschitz and linear growth conditions. In contrast to classical Lévy-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 Lévy 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 in the time-changed Lévy 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 Lévy equation to the corresponding time-changed problem. Finally, numerical experiments are presented to further support the theoretical findings.

Paper Structure

This paper contains 9 sections, 16 theorems, 150 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Assumptions and preliminary results
  3. Weak convergence order of stochastic theta method
  4. $p$th-moment boundedness of the stochastic theta approximation
  5. One-step weak expansion for the frozen Euler step
  6. Weak contribution of the implicit correction
  7. Weak convergence order of stochastic theta method
  8. Numerical experiments
  9. Conclusion

Key Result

Lemma 2.3

Suppose that Assumptions ass:globallinear and ass:techniquecond hold. Then we have the following conclusions:

Figures (3)

  • Figure 1: Sample paths of stochastic processes with $\Delta = 2^{-10}$ and $\theta = 0.5$ for \ref{['eq:onedimensional']}
  • Figure 2: Weak convergence order with test function $\Phi(x) = e^{-x^{2}}$ for \ref{['eq:onedimensional']}
  • Figure 3: Weak convergence order with test function $\Phi(x_{1},x_{2}) = x_{1}x_{2}$ for \ref{['eq:twodimensional']}

Theorems & Definitions (32)

  • Lemma 2.3
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 22 more