The truncated Euler–Maruyama method for stochastic differential equations

Influenced by Higham et al. (2003), 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. In this paper we will develop a new explicit method, called the truncated EM method, for the nonlinear SDE dx(t)=f(x(t))dt+g(x(t))dB(t)dx(t)=f(x(t))dt+g(x(t))dB(t) and establish the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition View the MathML sourcexTf(x)+p−12∣g(x)∣2≤K(1+∣x∣2). The type of convergence specifically addressed in this paper is strong-LqLq convergence for 2≤q<p2≤q<p, and pp is a parameter in the Khasminskii-type condition

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

Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients

In this work, we generalize the current theory of strong convergence rates for the backward Euler–Maruyama scheme for highly non-linear stochastic differential equations, which appear in both mathematical finance and bio-mathematics. More precisely, we show that under a dissipative condition on the drift coefficient and superlinear growth condition on the diffusion coefficient the BEM scheme converges with strong order of a half. This type of convergence gives theoretical foundations for efficient variance reduction techniques for Monte Carlo simulations. We support our theoretical results with relevant examples, such as stochastic population models and stochastic volatility models

Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients

We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods