Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations

Influenced by Higham, Mao and Stuart [9], several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations (SDEs) under the local Lipschitz condition. These numerical methods include the tamed Euler–Maruyama (EM) method, the tamed Milstein method, the stopped EM, the backward EM, the backward forward EM, etc. Recently, we developed a new explicit method in [23], called the truncated EM method, for the nonlinear SDE dx(t) = f (x(t))dt + g(x(t))dB(t) and established the strong convergence theory under the local Lip- schitz condition plus the Khasminskii-type condition xT f (x) + p−1 |g(x)|2 ≤ K(1 + |x|2). However, due to the page limit there, we did not study the convergence rates for the method, which is the aim of this paper. We will, under some additional conditions, discuss the rates of Lq -convergence of the truncated EM method for 2 ≤ q < p and show that the order of Lq -convergence can be arbitrarily close to q/2

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

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.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

Strong rate of convergence for the Euler-Maruyama approximation of SDEs with H\"older continuous drift coefficient

In this paper, we consider a numerical approximation of the stochastic differential equation (SDE) $$X_{t}=x_{0}+ \int_{0}^{t} b(s, X_{s}) \mathrm{d}s + L_{t},~x_{0} \in \mathbb{R}^{d},~t \in [0,T],$$ where the drift coefficient $b:[0,T] \times \mathbb{R}^d \to \mathbb{R}^d$ is H\"older continuous in both time and space variables and the noise $L=(L_t)_{0 \leq t \leq T}$ is a $d$-dimensional L\'evy process. We provide the rate of convergence for the Euler-Maruyama approximation when $L$ is a Wiener process or a truncated symmetric $\alpha$-stable process with $\alpha \in (1,2)$. Our technique is based on the regularity of the solution to the associated Kolmogorov equation.Comment: 19 page