A Simple Proof of Inequalities of Integrals of Composite Functions

In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continous convex functions on a vector space ${\bf R}^m$ and vector-valued functions in a weakly compact subset of a Banach vector space generated by $m$ $L_\mu^p$-spaces for $1\leq p<+\infty.$ Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by $m$ $L_\mu^\infty$-spaces instead

The semi-implicit Euler-Maruyama method for nonlinear non-autonomous stochastic differential equations driven by a class of L\'evy processes

The strong convergence of the semi-implicit Euler-Maruyama (EM) method for stochastic differential equations with non-linear coefficients driven by a class of L\'evy processes is investigated. The dependence of the convergence order of the numerical scheme on the parameters of the class of L\'evy processes is discovered, which is different from existing results. In addition, the existence and uniqueness of numerical invariant measure of the semi-implicit EM method is studied and its convergence to the underlying invariant measure is also proved. Numerical examples are provided to confirm our theoretical results.Comment: 16 pages, 6 figure

Block methods for linear Hamiltonian systems

For the numerical treatment of Hamiltonian differential equations,symplectic integrators are regarded as the most suitable choice.In this paper we are concerned with the applicability of block methods for the discrete approximate solution of linear Hamiltonian systems.The k-dimensional block methods are convergent of order at least k+1 for ordinary differential equations.We provide conditions on the coefficients of the equivalent block methods in order to maintain two important properties of linear Hamiltonian problems.It is shown that the k-dimensional block method which is convergent of order at least k+1 is symplectic and preserves the quadratic form at the last point of the block for k=1,2,…,8. Numerical experiment is given to illustrate the performance of the block methods

The Linear Relationship Model with LASSO for Studying Stock Networks

The correlation-based network is a powerful tool to reveal the influential mechanisms and relations in stock markets. However, current methods for developing network models are dominantly based on the pairwise relationship of positive correlations. This work proposes a new approach for developing stock relationship networks by using the linear relationship model with LASSO to explore negative correlations under a systemic framework. The developed model not only preserves positive links with statistical significance but also includes link directions and negative correlations. We also introduce blends cliques with the balance theory to investigate the consistency properties of the developed networks. The ASX 200 stock data with 194 stocks are applied to evaluate the effectiveness of our proposed method. Results suggest that the developed networks not only are highly consistent with the correlation coefficient in terms of positive or negative correlations but also provide influence directions in stock markets