Thesis

Advances in the Euler-Maruyama method for stochastic differential equations with local Lipschitz coefficients

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Awarding institution
  • University of Strathclyde
Date of award
  • 2025
Thesis identifier
  • T17392
Person Identifier (Local)
  • 202172333
Qualification Level
Qualification Name
Department, School or Faculty
Abstract
  • My PhD research is devoted to enriching the strong convergence theory of modified Euler-Maruyama methods for stochastic differential equation with locally Lipschitz coefficients. In this PhD thesis, we will introduce several modified Euler-Maruyama methods and establish their strong convergence theory. First, we will use new numerical analysis techniques to improve strong convergence results of the truncated Euler-Maruyama method. We then combine analysis techniques for polynomially growing coefficients and concave coefficients to extend the truncated EM method for multidimensional SDEs with polynomially growing drift and concave diffusion coefficients satisfying the Osgood condition. Then we will pay attention to scalar SDEs with locally Lipschitz coefficients. We will start with improving strong convergence results of the logarithmic truncated Euler-Maruyama method. To be concrete, we will use new numerical analysis techniques and further extend them for the constant elasticity of variance model and the A¨ıt-Sahalia model with almost full parameter ranges. We will prove that the logarithmic truncated Euler-Maruyama method is strongly convergent with order one half in general Lp-norm. In the rest of this thesis, we will focus on the projected Euler-Maruyama method. It has good convergence properties for scalar SDEs with locally Lipschitz coefficients. For example, it is strong Lp-convergent with order one half for the Cox-Ingersoll-Ross model with a wide parameter ranges. In particular, we will introduce a novel numerical analysis technique to prove that the projected Euler-Maruyama method may have finite inverse moments, which other modified Euler-Maruyama methods generally do not have. We will use finite inverse moments to prove that the projected Euler-Maruyama method is strong Lp - convergent with order one for many useful scalar SDE models, e.g., the constant elasticity of variance model, the A¨ıt-Sahalia model, the Heston-3/2 volatility model, the Wright-Fisher model and so on.
Advisor / supervisor
  • Mao, Xuerong
Resource Type
DOI
Date Created
  • 2024

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