The partially truncated Euler-Maruyama (EM) method is proposed in this paper for highly nonlinear stochastic differential equations (SDEs). We will not only establish the finite-time strong L r -convergence theory for the partially truncated EM method, but also demonstrate the real benefit of the method by showing that the method can preserve the asymptotic stability and boundedness of the underlying SDEs.

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao [15], and the theory there showed that the Euler-Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in L p ) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao [16] to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition. e.g., [3,7,10,13,18,19]). The numerical solutions of SDDEs under the generalized Khasminskiitype condition were discussed by Mao [15], and the theory there showed that the Euler-Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in L p ) of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.Recently, Mao [16] develops a new explicit numerical method, called the truncated EM method, for SDEs under the Khasminskii-type condition plus the local Lipschitz condition and establishes the strong convergence theory. In this paper, we will use this new truncated EM method to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.This paper is organized as follows: We will introduce necessary notion, state the generalized Khasminskii-type condition and define the truncated EM numerical solutions for SDDEs in Section 2. We will establish the strong convergence theory for the truncated EM numerical solutions in Sections 3 and 4 and discuss the convergence rates in Section 5. In each of these three sections we will illustrate our theory by examples. We will see from these examples that the truncated EM numerical method can be applied to approximate the solutions of many highly nonlinear SDDEs. We will finally conclude our paper in Section 6. The Truncated Euler-Maruyama MethodThroughout this paper, unless otherwise specified, we use the following notation. Let | · | be the Euclidean norm in R n . If A is a vector or matrix, its transpose is denoted by A T . If A is a matrix, its trace norm is denoted by |A| = trace(A T A). Let R + = [0, ∞) and τ > 0. Denote by C([−τ, 0]; R n ) the family of continuous functions from [−τ, 0] to R n with the norm ϕ = sup −τ ≤θ≤0 |ϕ(θ)|. Let (Ω, F , {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets). Let B(t) = (B 1 (t), · · · , B m (t)) T be an m-dimensional Brownian motion defined on the probability space. Moreover, for two real numbers a and b, we use a ∨ b = max(a, b) and a ∧ b = min(a, b). If G is a set, its indicator function is denoted by I G , namely I G (x) = 1 if x ∈ G and 0 otherwise. If a is a real number, we denote by ⌊a⌋ the largest inte...

SUMMARYBy a general Lagrange multiplier, an iteration approach is proposed to solve the generalized normalized diode equation, by suitable choice of the initial trial-function, one-step iteration leads to a high accurate solution, which is valid for the whole solution domain.

This paper is concerned with the almost sure exponential stability of the multidimensional nonlinear stochastic differential delay equation (SDDE) with variable delays of the form dx(t) = f (x(t − δ 1 (t)), t)dt + g(x(t − δ 2 (t)), t)dB(t), where δ 1 , δ 2 : R + → [0, τ ] stand for variable delays. We show that if the corresponding (nondelay) stochastic differential equation (SDE) dy(t) = f (y(t), t)dt + g(y(t), t)dB(t) admits a Lyapunov function (which in particular implies the almost sure exponential stability of the SDE) then there exists a positive number τ * such that the SDDE is also almost sure exponentially stable as long as the delay is bounded by τ *. We provide an implicit lower bound for τ * which can be computed numerically. Moreover, our new theory enables us to design stochastic delay feedback controls in order to stabilize unstable differential equations.

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