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.
Multi-arm trials meta-analysis is a methodology used in combining evidence based on a synthesis of different types of comparisons from all possible similar studies and to draw inferences about the effectiveness of multiple compared-treatments. Studies with statistically significant results are potentially more likely to be submitted and selected than studies with non-significant results; this leads to false-positive results. In meta-analysis, combining only the identified selected studies uncritically may lead to an incorrect, usually over-optimistic conclusion. This problem is known asbiselection bias. In this paper, we first define a random-effect meta-analysis model for multi-arm trials by allowing for heterogeneity among studies. This general model is based on a normal approximation for empirical log-odds ratio. We then address the problem of publication bias by using a sensitivity analysis and by defining a selection model to the available data of a meta-analysis. This method allows for different amounts of selection bias and helps to investigate how sensitive the main interest parameter is when compared with the estimates of the standard model. Throughout the paper, we use binary data from Antiplatelet therapy in maintaining vascular patency of patients to illustrate the methods.
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...
Abstract. The discrepancy arises in the worst-case error analysis for quasi-Monte Carlo quadrature rules. Low discrepancy sets yield good quadrature rules. This article considers the mean square discrepancies for scrambled (λ, t, m, s)-nets and (t, s)-sequences in base b. It is found that the mean square discrepancy for scrambled nets and sequences is never more than a constant multiple of that under simple Monte Carlo sampling. If the reproducing kernel defining the discrepancy satisfies a Lipschitz condition with respect to one of its variables separately, then the asymptotic order of the root mean square discrepancy is O(n −1 [log n] (s−1)/2 ) for scrambled nets. If the reproducing kernel satisfies a Lipschitz condition with respect to both of its variables, then the asymptotic order of the root mean square discrepancy is O(n −3/2 [log n] (s−1)/2 ) for scrambled nets. For an arbitrary number of points taken from a (t, s)-sequence, the root mean square discrepancy appears to be no better than O(n −1 [log n] (s−1)/2 ), regardless of the smoothness of the reproducing kernel.
Inspired by the truncated Euler-Maruyama method developed in Mao (J. Comput.Appl. Math. 2015), we propose the truncated Milstein method in this paper. The strong convergence rate is proved to be close to 1 for a class of highly non-linear stochastic differential equations. Numerical examples are given to illustrate the theoretical results.
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