[Paper Review] Convergence rates of truncated EM scheme for NSDDEs

Li Tan, Chenggui Yuan|arXiv (Cornell University)|Jan 18, 2018

Numerical methods for differential equations6 references3 citations

TL;DR

This paper establishes convergence rates for a truncated Euler-Maruyama (EM) scheme applied to neutral stochastic differential delay equations (NSDDEs) driven by Brownian motion and pure jump processes. Under local Lipschitz and one-sided Lipschitz conditions, along with a Khasminskii-type growth condition, the authors prove strong convergence with rate $ O(\Delta^{1/2} \overline{g}(\Delta)^p) $, where $ \overline{g}(\Delta) $ controls the truncation threshold, offering a robust numerical method for NSDDEs with non-globally Lipschitz coefficients.

ABSTRACT

This paper is concerned with strong convergence of the truncated Euler-Maruyama scheme for neutral stochastic differential delay equations driven by Brownian motion and pure jumps respectively. Under local Lipschitz condition, convergence rates of the truncated EM scheme are given.

Motivation & Objective

• To develop a numerically stable and convergent scheme for neutral stochastic differential delay equations (NSDDEs) with non-globally Lipschitz coefficients.
• To extend the truncated EM method—previously used for SDEs—to NSDDEs driven by Brownian motion and pure jump processes.
• To derive explicit convergence rates under local Lipschitz and one-sided Lipschitz conditions, including a Khasminskii-type growth condition.
• To ensure the numerical solution remains moment-bounded and avoids explosion in finite time, even under superlinear growth.

Proposed method

• Proposes a truncated EM scheme by modifying drift and diffusion coefficients using a truncation function $ f^{-1}(g(\Delta)) $, ensuring boundedness of coefficients for large values.
• Implements a stopping time $ \tau_R $ to control the growth of the exact solution and uses moment estimates to bound the probability of large deviations.
• Applies Itô's formula and Gronwall-type inequalities to derive moment bounds for both the exact and numerical solutions.
• Introduces a function $ \overline{g}(\Delta) $ that controls the truncation threshold and appears in the final convergence rate estimate.
• Uses assumptions (A1)–(A3) to ensure existence, uniqueness, and moment stability of the exact solution.
• Applies the Khasminskii-type condition (A3) to control the growth of the drift and diffusion terms, enabling convergence under local Lipschitz conditions.

Experimental results

Research questions

• RQ1 What is the strong convergence rate of the truncated EM scheme for NSDDEs driven by Brownian motion under local Lipschitz and Khasminskii-type conditions?
• RQ2 How does the convergence rate depend on the truncation function $ g(\Delta) $, and can it be quantified in terms of step size $ \Delta $?
• RQ3 Can the truncated EM scheme maintain stability and convergence for NSDDEs with superlinearly growing drift and diffusion coefficients?
• RQ4 How does the scheme perform for NSDDEs driven by pure jump processes, particularly under one-sided Lipschitz and superlinear growth conditions?
• RQ5 What is the role of the function $ \overline{g}(\Delta) $ in balancing truncation and convergence rate?

Key findings

• The truncated EM scheme achieves strong convergence with rate $ O(\Delta^{1/2} \overline{g}(\Delta)^p) $ for NSDDEs driven by Brownian motion under local Lipschitz and Khasminskii-type conditions.
• The convergence rate is explicitly dependent on the truncation function $ \overline{g}(\Delta) $, which grows as $ \Delta \to 0 $, reflecting the trade-off between stability and accuracy.
• For the case $ D(y) = \frac{1}{2}\sin y $, $ b(x,y) = x - x^3 + \cos y $, $ \sigma(x,y) = |x|^{3/2} $, the conditions (A1)–(A4) are satisfied, validating the theoretical framework.
• The exact solution satisfies $ \sup_{0 \leq t \leq T} \mathbb{E}|X(t)|^p \leq C $, and $ \mathbb{P}(\tau_R \leq T) \leq C / R^p $, ensuring moment stability.
• The truncated scheme preserves the moment bounds of the exact solution, with the numerical solution satisfying $ \mathbb{E}(\sup_{0 \leq t \leq T} |X(t) - Y(t)|^p) \leq C \Delta^{1/2} \overline{g}(\Delta)^p $.
• The analysis confirms that the truncated EM scheme is effective for NSDDEs with non-globally Lipschitz coefficients, including those with superlinear growth and one-sided Lipschitz drifts.