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No risk assessment tools for the efficacy of folic acid treatment for hyperhomocysteinaemia (HHcy) have been developed. We aimed to use two common genetic risk score (GRS) methods to construct prediction models for the efficacy of folic acid therapy on HHcy, and the best gene–environment prediction model was screened out. A prospective cohort study enrolling 638 HHcy patients was performed. We used a logistic regression model to estimate the associations of two GRS methods with the efficacy. Performances were compared using area under the receiver operating characteristic curve (AUC). The simple count genetic risk score (SC-GRS) and weighted genetic risk score (wGRS) were found to be independently associated with the efficacy of folic acid treatment for HHcy. Using the SC-GRS, per risk allele increased with a 1·46-fold increased failure risk (P < 0·001) after adjustment for traditional risk factors, including age, sex, BMI, smoking, alcohol consumption, history of diabetes, history of hypertension, history of hyperlipidaemia, history of stroke and history of CHD. When used the wGRS, the association was strengthened (OR = 2·08, P < 0·001). Addition of the SC-GRS and wGRS to the traditional risk model significantly improved the predictive ability by AUC (0·859). A precise gene–environment predictive model with good performance was developed for predicting the treatment failure rate of folic acid therapy for HHcy.
This paper considers the effect of the biases in Global Positioning System (GPS) observations on satellite clock offset estimation. GPS triple-frequency satellite clock and reference observations are discussed. When the reference observation is selected and the corresponding satellite clock offset is computed, satellite clock offsets for all observations are obtained based on the computed satellite clock offset and the biases between the reference observation and other observations. The characteristics of these biases are analysed, and a service strategy for the GPS triple-frequency satellite clock offset is presented. To evaluate the computed GPS satellite clock offset, the performance in single-point positioning is validated. The positioning results show that the average relative improvements are about 20%, 28% and 19% for north, east and vertical components, when the Differential Code Bias (DCB) (P1-P2), DCB (P1-P5) and modelled Inter-Frequency Clock Bias (IFCB) are corrected. The effect of DCB (P1-P2), DCB (P1-P5) and modelled IFCB on the altitude direction is more evident than on the horizontal directions.
This paper is devoted to numerical methods for mean-field stochastic differential equations (MSDEs). We first develop the mean-field Itô formula and mean-field Itô-Taylor expansion. Then based on the new formula and expansion, we propose the Itô-Taylor schemes of strong order γ and weak order η for MSDEs, and theoretically obtain the convergence rate γ of the strong Itô-Taylor scheme, which can be seen as an extension of the well-known fundamental strong convergence theorem to the mean-field SDE setting. Finally some numerical examples are given to verify our theoretical results.
In error estimates of various numerical approaches for solving decoupled forward backward stochastic differential equations (FBSDEs), the rate of convergence for one variable is usually less than for the other. Under slightly strengthened smoothness assumptions, we show that the fully discrete Euler scheme admits a first-order rate of convergence for both variables.
The deferred correction (DC) method is a classical method for solving ordinary differential equations; one of its key features is to iteratively use lower order numerical methods so that high-order numerical scheme can be obtained. The main advantage of the DC approach is its simplicity and robustness. In this paper, the DC idea will be adopted to solve forward backward stochastic differential equations (FBSDEs) which have practical importance in many applications. Noted that it is difficult to design high-order and relatively “clean” numerical schemes for FBSDEs due to the involvement of randomness and the coupling of the FSDEs and BSDEs. This paper will describe how to use the simplest Euler method in each DC step–leading to simple computational complexity–to achieve high order rate of convergence.
This is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve the second-order FBSDEs (2FBSDEs). The key feature of the multistep schemes is that the Euler method is used to discretize the forward SDE, which dramatically reduces the entire computational complexity. Moreover, it is shown that the usual quantities of interest (e.g., the solution tuple (Yt,Zt,At,Γt) of the 2FBSDEs) are still of high order accuracy. Several numerical examples are given to show the effectiveness of the proposed numerical schemes. Applications of our numerical schemes to stochastic optimal control problems are also presented.
In this paper, we investigate the mean-square convergence of the split-step θ-scheme for nonlinear stochastic differential equations with jumps. Under some standard assumptions, we rigorously prove that the strong rate of convergence of the split-step θ-scheme in strong sense is one half. Some numerical experiments are carried out to assert our theoretical result.
By introducing a new Gaussian process and a new compensated Poisson random measure, we propose an explicit prediction-correction scheme for solving decoupled forward backward stochastic differential equations with jumps (FBSDEJs). For this scheme, we first theoretically obtain a general error estimate result, which implies that the scheme is stable. Then using this result, we rigorously prove that the accuracy of the explicit scheme can be of second order. Finally, we carry out some numerical experiments to verify our theoretical results.
Upon a set of backward orthogonal polynomials, we propose a novel multi-step numerical scheme for solving the decoupled forward-backward stochastic differential equations (FBSDEs). Under Lipschtiz conditions on the coefficients of the FBSDEs, we first get a general error estimate result which implies zero-stability of the proposed scheme, and then we further prove that the convergence rate of the scheme can be of high order for Markovian FBSDEs. Some numerical experiments are presented to demonstrate the accuracy of the proposed multi-step scheme and to numerically verify the theoretical results.
Convergence analysis is presented for recently proposed multistep schemes, when applied to a special type of forward-backward stochastic differential equations (FB-SDEs) that arises in finance and stochastic control. The corresponding k-step scheme admits a k-order convergence rate in time, when the exact solution of the forward stochastic differential equation (SDE) is given. Our analysis assumes that the terminal conditions and the FBSDE coefficients are sufficiently regular.
In this paper, we are concerned with probabilistic high order numerical schemes
for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs,
it is shown by Cheridito, Soner, Touzi and Victoir  that the associated exact
solutions admit probabilistic interpretations, i.e., the solution of a fully
nonlinear parabolic PDE solves a corresponding second order forward backward
stochastic differential equation (2FBSDEs). Our numerical schemes rely on
solving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T.
Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.]. Moreover, in our
numerical schemes, one has the flexibility to choose the associated forward SDE,
and a suitable choice can significantly reduce the computational complexity.
Various numerical examples including the HJB equations are presented to show the
effectiveness and accuracy of the proposed numerical schemes.
An explicit numerical scheme is proposed for solving decoupled forward backward stochastic differential equations (FBSDE) represented in integral equation form. A general error inequality is derived for this numerical scheme, which also implies its stability. Error estimates are given based on this inequality, showing that the explicit scheme can be second-order. Some numerical experiments are carried out to illustrate the high accuracy of the proposed scheme.
In this paper, a new numerical method for solving the decoupled forward-backward stochastic differential equations (FBSDEs) is proposed based on some specially derived reference equations. We rigorously analyze errors of the proposed method under general situations. Then we present error estimates for each of the specific cases when some classical numerical schemes for solving the forward SDE are taken in the method; in particular, we prove that the proposed method is second-order accurate if used together with the order-2.0 weak Taylor scheme for the SDE. Some examples are also given to numerically demonstrate the accuracy of the proposed method and verify the theoretical results.
Gongyi is the birthplace and one of the most famous production areas of white porcelain in China. In this study, white porcelain samples from the Northern Wei to Tang Dynasties excavated from Baihe and Huangye kiln sites were analyzed to investigate microstructure and its physicochemical basis. The result demonstrates that the formation of an interaction layer of Anorthite crystals and the accompanied phase-separation structure at glaze-body boundary is a common character of microstructure. And there is little probability for crystal precipitation within the glaze layer.
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