Thesis

Truncated Euler Maruyama numerical method for stochastic differential (delay) equations models

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Awarding institution
  • University of Strathclyde
Date of award
  • 2024
Thesis identifier
  • T16813
Person Identifier (Local)
  • 201967364
Qualification Level
Qualification Name
Department, School or Faculty
Abstract
  • In this thesis, our focus has been on enhancing the applicability and reliability of the truncated Euler-Maruyama (EM) numerical method for stochastic differential equations (SDEs) and stochastic delay differential equations (SDDEs), initially introduced by Mao [21]. Building upon this method, our contributions span several chapters. In Chapter 3, we pointed out its limitations in determining the convergence rate over a finite time interval and established a new result for SDEs whose diffusion coefficients may not satisfy the global Lipschitz condition. We extended our exploration to include time delays in Chapter 4, allowing for varying delays over time. The chapter also introduces additional lemmas to ensure the convergence rates of the method to the solution at specific time points and over finite intervals. However, the global Lipschitz condition on the diffusion coefficient is currently required. In Chapter 5, we focused on the Lotka-Volterra model, introducing modifications such as the Positive Preserving Truncated EM (PPTEM) and Nonnegative Preserving Truncated EM (NPTEM) methods to handle instances where the truncated EM method generated nonsensical negative solutions. The proposed adjustments, guided by Assumption 5.1.1, ensure that the numerical solutions remain meaningful and interpretable. Chapter 6 extends these concepts to the stochastic delay Lotka-Volterra model with a variable time delay, demonstrating the adaptability and applicability of our methods. Despite we also assume the stronger condition 6.1.1 to prove the convergence of numerical solutions, future research aims to explore relaxed conditions, broadening the applicability of these numerical methods. Overall, this thesis contributes to establishing convergence rates for SDEs under local Lipschitz diffusion coefficients, extending the methodology to address time delays and modifying the truncated EM method to ensure positive and nonnegative numerical solutions. These advancements are demonstrated through applications to the stochastic variable time delay Lotka-Volterra model, emphasizing the meaningfulness and interpretability of the solutions.
Advisor / supervisor
  • Mao, Xuerong
Resource Type
DOI
Date Created
  • 2023
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