[Paper Review] Stability of analytical solutions and convergence of numerical methods for non-linear stochastic pantograph differential equations

Minghui Song, Yulan Lu+1 more|arXiv (Cornell University)|Jan 31, 2015

Stochastic processes and financial applications8 references3 citations

TL;DR

This paper establishes sufficient conditions for polynomial stability of analytical solutions and proves convergence of the semi-implicit Euler method for non-linear stochastic pantograph differential equations (SPDEs). It derives mean-square and almost surely polynomial stability under Lipschitz-type conditions, and proves second-order consistency and convergence in mean-square and average senses, with convergence order $ O(h^{1/2}) $, under appropriate assumptions on the drift and diffusion coefficients.

ABSTRACT

In this paper, we study the polynomial stability of analytical solution and convergence of the semi-implicit Euler method for non-linear stochastic pantograph differential equations. Firstly, the sufficient conditions for solutions to grow at a polynomial rate in the sense of mean-square and almost surely are obtained. Secondly, the consistence and convergence of this method are proved. Furthermore, the orders of consistence (in the sense of average and mean-square) and convergence are given, respectively.

Motivation & Objective

• To analyze the polynomial stability of analytical solutions for non-linear stochastic pantograph differential equations (SPDEs).
• To establish sufficient conditions under which solutions grow at a polynomial rate in the mean-square and almost surely sense.
• To investigate the consistency and convergence of the semi-implicit Euler method for SPDEs with pantograph-type delays.
• To derive the orders of consistency and convergence in both mean-square and average senses for the numerical method.

Proposed method

• Derives sufficient conditions for polynomial stability using Lyapunov-type functionals and comparison principles for stochastic differential equations.
• Applies the semi-implicit Euler method to discretize the SPDE, with a parameter $ \theta \in [0,1] $ controlling implicitness.
• Uses Itô's formula and moment estimates to bound the error between the exact and numerical solutions.
• Establishes error bounds via recursive inequalities involving conditional expectations and martingale properties.
• Applies Gronwall-type arguments and stochastic estimates to control the propagation of error over time.
• Derives convergence order $ O(h^{1/2}) $ in mean-square sense by bounding the maximum error over the time grid.

Experimental results

Research questions

• RQ1 Under what conditions do solutions of non-linear SPDEs with pantograph delays exhibit polynomial stability in the mean-square sense?
• RQ2 What are the sufficient conditions for almost sure polynomial stability of the zero solution in such SPDEs?
• RQ3 How does the semi-implicit Euler method perform in terms of consistency and convergence for these equations?
• RQ4 What is the order of consistency and convergence of the semi-implicit Euler method in mean-square and average senses?
• RQ5 How do the drift and diffusion coefficients' Lipschitz properties affect the convergence behavior?

Key findings

• The zero solution is mean-square polynomial stable if the drift and diffusion coefficients satisfy specific Lipschitz conditions and the system parameters satisfy $ \bar{a} + \bar{b} q^\alpha = 0 $ with $ \alpha < 0 $.
• Almost sure polynomial stability holds under the same Lipschitz conditions, with the solution decaying at a rate controlled by $ t^\alpha $ almost surely.
• The semi-implicit Euler method is consistent with order $ O(h) $ in the average sense and order $ O(h^{1/2}) $ in the mean-square sense.
• The method is convergent with order $ O(h^{1/2}) $ in the mean-square sense, uniformly over the time interval $[0,T]$.
• The convergence result holds for $ \theta \in [0,1] $, with the error bound independent of the delay ratio $ q \in (0,1) $, provided $ h $ is sufficiently small.
• The error estimate is uniform in time and depends on the initial condition and the Lipschitz constants of the coefficients.