SEQUENTIAL QUADRATIC PROGRAMMING METHODS FOR OPTIMAL CONTROL PROBLEMS WITH STATE CONSTRAINTS

A Kind of direct methods is presented for the solution of optimal control problems with state constraints.These methods are sequential quadratic programming methods.At every iteration a quadratic programming which is obtained by quadratic approximation to Lagrangian function and Linear approximations to constraints is solved to get a search direction for a merit function.The merit function is formulated by augmenting the Lagrangian funetion with a penalty term.A line search is carried out along the search direction to determine a step length such that the merit function is decreased.The methods presented in this paper include continuous sequential quadratic programming methods and discreate sequential quadrade programming methods.     

Fourth-Order Splitting Methods for Time-Dependant Differential Equations

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes,e.g.,wave-propagation or heat-transfer,that are modeled by wave equations or heat equations.Here,we study both parabolic and hyperbolic equations.We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods,which are standard splitting methods of lower order,e.g.second-order.Our aim is to develop higher-order ADI methods,which are performed by Richardson extrapolation,Crank-Nicolson methods and higher-order LOD methods,based on locally higher-order methods.We discuss the new theoretical results of the stability and consistency of the ADI methods.The main idea is to apply a higher- order time discretization and combine it with the ADI methods.We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives.The higher-order methods are unconditionally stable.Some numerical experiments verify our results.     

具有非单调出生函数的周期反应扩散模型的渐近传播速度(英文)

A reaction-diffusion model for a single species with age structure and nonlocal reaction for periodic time t is derived. Some results about the model with monotone birth function are firstly introduced, and then by constructing two auxiliary equations and squeezing method, the spreading speed for the system with nonmonotone birth function is obtained.     

求解无约束非线性规划问题的一个新的重开始共轭梯度算法的收敛性

Conjugate gradient optimization algorithms depend on the search directions.with different choices for the parameters in the search directions.In this note,by combining the nice numerical performance of PR and HS methods with the global convergence property of the class of conjugate gradient methods presented by HU and STOREY(1991),a class of new restarting conjugate gradient methods is presented.Global convergences of the new method with two kinds of common line searches,are proved .Firstly,it is shown that,using reverse modulus of continuity funciton and forcing function,the new method for solving unconstrained optimization can work for a continously differentiable function with Curry-Altman‘s step size rule and a bounded level set .Secondly,by using comparing technique,some general convergence propecties of the new method with other kind of step size rule are established,Numerical experiments show that the new method is efficient by comparing with FR conjugate gradient method.     

IMPLICIT-EXPLICIT MULTISTEP FINITE ELEMENT-MIXED FINITE ELEMENT METHODS FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE

The transient behavior of a semiconductor device consists of a Poisson equa-tion for the electric potential and of two nonlinear parabolic equations for the electrondensity and hole density.The electric potential equation is discretized by a mixed finiteelement method. The electron and hole density equations are treated by implicit-explicitmultistep finite element methods. The schemes are very efficient. The optimal order errorestimates both in time and space are derived.     

对于金融资产波动率估计的双域组合方法

Time- and state-domain methods are two common approaches for nonparametrically estimating the volatility of financial assets. Economic conditions vary over time in real financial market. It is reasonable to expect that volatility depends on both time and price level for a given state variable. Recently, Fan, et al （2007） proposed the idea of dynamically integrated method in both time-and state domain. This idea has become an interesting topic in the estimation of volatility. In this paper, our purpose is to discuss the integrated method in the estimation of volatility. Simulations are conducted to demonstrate that the newly integrated method outperforms some old ones, and the results of simulations demonstrate this fact. Furthermore, we establish its asymptotic properties.     

The Uniform <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Behavior for Time Discretization of an Evolution Equation

The uniform L^2 stability and convergence properties for the time discretization of an evolution equation with a memory term are studied.The methods are based on the second-order backward difference methods.The memory term is approximated by the second-order convolution quadrature and interpolant quadrature.     

关于错排数与Bell数的一些循环公式

In this paper, a further investigation for the number of Derangements and Bell numbers is performed, and some new recursion formulae for the number of Derangements and Bell numbers are established by applying the generating function methods and Padé approximation techniques. Illustrative special cases of the main results are also presented.     

A TRUST REGION METHOD WITH A CONIC MODEL FOR NONLINEARLY CONSTRAINED OPTIMIZATION

Trust region methods are powerful and effective optimization methods.The conic model method is a new type of method with more information available at each iteration than standard quadratic-based methods.The advantages of the above two methods can be combined to form a more powerful method for constrained optimization.The trust region subproblem of our method is to minimize a conic function subject to the linearized constraints and trust region bound.At the same time,the new algorithm still possesses robust global properties.The global convergence of the new algorithm under standard conditions is established.     

结合修正Curry-Altman步长搜索一个新的共轭梯度算法的收敛性

Conjugate gradient optimization algorithms depend on the search directions with different choices for the parameter in the search directions. In this note, conditions are given on the parameter in the conjugate gradient directions to ensure the descent property of the search directions. Global convergence of such a class of methods is discussed. It is shown that, using reverse modulus of continuity function and forcing function, the new method for solving unconstrained optimization can work for a continuously differentiable function with a modification of the Curry-Altman‘s step-size rule and a bounded level set. Combining PR method with our new method, PR method is modified to have global convergence property.Numerical experiments show that the new methods are efficient by comparing with FR conjugate gradient method.     

含噪音高频数据动态整合估计波动率的方法

时间域和状态域方法是两种常见的非参数估计方法.前者主要使用的是最近的历史数据,而后者则主要依赖于过去的历史信息.本文在时间域上,通过对含噪音高频数据采用双时间尺度方法获得其波动率,进而获得经动态整合后的波动率.     

扩散模型中扩散系数的非参数组合估计

为了提高扩散系数估计的准确度, 我们利用动态组合时间域与状态域信息提出一个新的组合估计量. 我们发现所提组合估计量能有效估计扩散模型的扩散系数, 正如在本文中模拟所示. 在一定的条件下, 建立了估计量的渐进正态性, 并证明了时间域估计量与状态域估计量是渐进独立的. 大量的模拟展示了所提组合估计量优于单域估计量, 也优于本文所提估计量.     

A class of split-step balanced methods for stiff stochastic differential equations

In this paper we design a class of general split-step balanced methods for solving It? stochastic differential systems with m-dimensional multiplicative noise, in which the drift or deterministic increment function can be taken from any chosen one-step ODE solver. We then give an analysis of their order of strong convergence in a general setting, but for the mean-square stability analysis, we confine our investigation to a special case in which the drift increment function of the methods is replaced by the one from the well known Rosenbrock method. The resulting class of stochastic differential equation (SDE) solvers will have more appropriate and useful mean-square stability properties for SDEs with stiffness in their drift and diffusion parts, compared to some other already reported split-step balanced methods. Finally, numerical results show the effectiveness of these methods.     

Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise

We consider estimation of the drift function of a stationary diffusion process when we observe high-frequency data with microstructure noise over a long time interval. We propose to estimate the drift function at a point by a Nadaraya–Watson estimator that uses observations that have been pre-averaged to reduce the noise. We give conditions under which our estimator is consistent and asympotically normal. Its rate and asymptotic bias and variance are the same as those without microstructure noise. To use our method in data analysis, we propose a data-based cross-validation method to determine the bandwidth in the Nadaraya–Watson estimator. Via simulation, we study several methods of bandwidth choices, and compare our estimator to several existing estimators. In terms of mean squared error, our new estimator outperforms existing estimators.     

随机微分方程改进的分裂步单支θ方法的强收敛性

本文提出了一个改进的分裂步单支θ方法,在漂移项系数满足单边Lipschitz条件下,证明了当数值方法的参数θ满足1/2 ≤ θ ≤ 1时,该数值方法对于这类随机微分方程是强收敛的,并在现有文献的基础上将方法的收敛阶从1/2阶提高到1阶;当0 ≤ θ ≤ 1/2时,若漂移项系数进一步满足线性增长条件,该数值方法也是强收敛的,收敛阶为1阶.文末的数值试验验证了理论结果的正确性.     

Runge-Kutta methods for jump-diffusion differential equations

In this paper we consider Runge-Kutta methods for jump-diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge-Kutta methods. First, we analyse schemes where the drift is approximated by a Runge-Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge-Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge-Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings.     

Optimal expulsion and optimal confinement of a Brownian particle with a switching cost

We solve two stochastic control problems in which a player tries to minimize or maximize the exit time from an interval of a Brownian particle, by controlling its drift. The player can change from one drift to another but is subject to a switching cost. In each problem, the value function is written as the solution of a free boundary problem involving second order ordinary differential equations, in which the unknown boundaries are found by applying the principle of smooth fit. For both problems, we compute the value function, we exhibit the optimal strategy and we prove its generic uniqueness.     

Runge–Kutta methods for jump–diffusion differential equations

In this paper we consider Runge–Kutta methods for jump–diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge–Kutta methods. First, we analyse schemes where the drift is approximated by a Runge–Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge–Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge–Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings.     

The tail empirical process of regularly varying functions of geometrically ergodic Markov chains

We consider a stationary regularly varying time series which can be expressed as a function of a geometrically ergodic Markov chain. We obtain practical conditions for the weak convergence of the tail array sums and feasible estimators of cluster statistics. These conditions include the so-called geometric drift or Foster–Lyapunov condition and can be easily checked for most usual time series models with a Markovian structure. We illustrate these conditions on several models and statistical applications. A counterexample is given to show a different limiting behavior when the geometric drift condition is not fulfilled.