不确定微分方程的数值解法及其应用研究

不确定微分方程广泛应用于不确定财政、不确定控制、不确定微分博弈等领域。由于一些不确定微分方程解析解难以实现,本文首先研究了不确定微分方程的Euler方法和Runge-Kutta方法两种数值解法,并进行误差分析。通过比较随机领域Black-Scholes模型和不确定领域Liu模型的欧式期权定价公式,验证不确定微分方程描述证券市场的合理性和实用性。     

浮动利率下基于不确定分数阶微分方程的期权定价研究

期权定价是金融数学领域中最复杂的问题之一.随着不确定理论公理化的建立,利用不确定理论进行期权定价的研究逐步展开,而分数阶微分方程的分数阶导数项可以很好地刻画金融市场的记忆特性.本文在机会空间中提出了一种新的不确定市场模型,假设股票价格满足Caputo型的不确定分数阶微分方程,且随机利率满足随机微分方程.基于该模型,利用Mittag-Leffler函数和微分方程的α-轨道我们给出了蝶式期权和欧式价差期权的定价公式及数值例子.     

具有时滞和参数不确定性的网络系统的滑模控制

研究了一类具有时滞的不确定网络控制系统(NCS)的滑模控制.首先,用一种特殊的变换将不确定时滞系统化为非延迟不确定系统形式.然后基于Lyapunov稳定性理论及线性矩阵不等式(LMI)技术,以线性矩阵不等式的形式给出了该系统的一些新的稳定性判据.所提出的控制器能够使具有不确定性的时滞系统渐近稳定性,而且控制器结构简单,计算方便.最后通过数值模拟,验证了所设计方法的有效性.     

用不确定德尔菲法预测GDP

德尔菲法是一种建立在专家意见基础上的预测评估方法.不确定统计是利用不确定理论收集和整理分析专家数据的一种统计方法,其中关键的一点是如何构造不确定变量的不确定分布.把德尔菲法和不确定统计相结合,就得到了一种估计不确定分布的新方法——不确定德尔菲法.对该方法的估计误差进行了改进,得到了一种预测GDP的新方法,并利用其预测邯郸市的生产总值(GDP).     

模型不确定非线性随机系统的鲁棒性能准则设计

对于一类模型不确定非线性随机系统,用耗散性的观点发展了鲁棒性能准则理论.特别地,将确定性非线性系统理论中的耗散性概念引入到模型不确定随机非线性系统中,并以此作为基础来发展H∞理论.在精确模型随机非线性系统H∞基础上,建立了模型不确定系统L2增益和HJI不等式的可解性的关系.由于HJI偏微分方程难于求解,考虑模型参数满足某种适当匹配条件的系统的鲁棒性能准则问题,我们不需要通过求解HJI方程就可以得到此类系统的H∞控制律.     

不确定理论下的DC型养老金的最优投资策略问题

在不确定理论的框架下,研究确定缴费(DC)型养老金的最优投资策略问题.以最小化二次损失函数为目标,分别在固定缴费和不确定缴费的情形下,建立养老金的最优化模型.利用不确定动态规划法,证明了不确定最优性原理,得出了不确定最优性方程,通过求解不确定最优性方程得到最优给付率和最优投资策略.     

带干扰的Erlang(2)风险模型的不破产概率

本文讨论了带干扰的Erlang(2)风险模型,通过构造一个延迟更新过程,我们得到了不破产概率满足的积分-微分方程,进而得到了不破产概率的明确表达式．     

不确定三重交替更新过程

在更新过程中,交替更新过程是常见的重要的过程之一.本文首先介绍不确定三重交替更新过程,其中间隔时间都是不确定变量.然后证明了关于平均工作率的极限值的不确定三重交替更新过程定理.最后给出了一个实际应用例子.     

基于不确定指数O-U过程带有浮动利率模型的亚式期权定价

不确定金融是不确定理论在现代金融领域的一种应用,在解决金融问题中发挥着越来越重要的作用。而利率是一个重要的经济指标,经常受到一些不确定因素的影响,在研究期权定价时,有必要考虑浮动利率。本文提出了一种新的不确定指数Ornstein-Uhlenbeck过程模型,假设利率服从不确定均值回复过程,研究了期权定价问题,运用α-轨道方法,分别推导了亚式看涨期权和看跌期权定价公式。最后,设计了计算期权价格的数值算法,并给出数值算例。     

不规则星系中星体随机运动的几率密度与天体不确定关系

在这篇论文中，作者用Langevin随机微分方程来描写不规则星系中任何一个作准布朗型运动的星体的随机运动．并用相空间中的表示点（X．X）来描写其每一随机态.用Fokker－Planck方程计算出了这些随机态的稳态几率密度fs（X．X）；同时，用此fs（X．X）计算出了随机运动星体的速度和位置的方差与涨落．进而作者发现：随机运动的星体满足一种天体的不确定关系．     

Continuous dependence theorems on solutions of uncertain differential equations

In ordinary differential equation (ODE) and stochastic differential equation (SDE), the solution continuously depends on initial value and parameter under some conditions. This paper investigates the analogous continuous dependence theorems in uncertain differential equation (UDE). It proves two continuous dependence theorems, a basic one and a general one.     

Option pricing formulas based on uncertain fractional differential equation

Uncertain fractional differential equations have been playing an important role in modelling complex dynamic systems. Early researchers have presented the extreme value theorems and time integral theorem on uncertain fractional differential equation. As applications of these theorems, this paper investigates the pricing problems of American option and Asian option under uncertain financial markets based on uncertain fractional differential equations. Then the analytical solutions and numerical solutions of these option prices are derived, respectively. Finally, some numerical experiments are performed to verify the effectiveness of our results.

     

Decentralized robust control design for uncertain delay systems

We consider a class of large-scale uncertain delay systems. The uncertain parameter vector in the system is possibly fast time-varying. It may be nonlinear in the system dynamics. No statistical or fuzzy information of the uncertainty is known. Based on only the possible bound of the uncertain parameter, a decentralized linear robust control is proposed, which renders the system asymptotically stable.     

一类离散不确定线性时滞系统的鲁棒镇定

研究具有时变不确定参数的离散线性时滞系统的鲁棒控制问题,其中不确定性满足匹配条件,利用Ｌｙａｐｕｎｏｖ确定性理论,提出了鲁棒稳定性控制器一种新的设计方法,得到了这类离散不确定线性时滞系统可鲁棒镇定的充分条件。     

Perron type theorems for functional differential equations with infinite delay in a Banach space

For functional differential equations in a Banach space, we obtain some results on the asymptotic behavior of solutions, which correspond to Perron type theorems for differential equations without delay or with finite delay.     

Two families of liapunov functions for functional differential systems

Two families of Liapunov functions are employed to study the global stability and boundedness of functional differential systems. New stability and boundedness theorems are obtained. Applications of these theorems to some nonlinear differential systems with infinite delay are discussed. Project supported by the National Science Foundation of China Under Grants 69871005     

OSCILLATION RESULTS FOR A SECOND ORDER NEUTRAL DELAY DIFFERENTIAL EQUATIONS

Some new oscillation criteria are established for a second order neutral delay differential equations. These results improve oscillation results of Y.V. Rogo-vchenko for the retarded delay differential equations. The relevance of our theorems is illustrated with two carefully selected examples.     

Note on the Ladas' paper on oscillation and asymptotic behavior of solutions of differential equations with retarded argument

This paper consists of three theorems. The first theorem gives the asymptotic behavior of solutions of the nonlinear delay differential equation. The other theorems give sufficient conditions for oscillatory solutions of variants of that equation.     

Stability of impulsive functional differential equations via the Liapunov functional

In this work, we investigate the stability of a class of impulsive functional differential equations. Some general stability theorems are obtained. Our results can be applied to finite delay impulsive systems or infinite delay impulsive systems or impulsive systems involving both finite and infinite delays, in a unified way. Examples are also given to illustrate that applying our theorems yields better conclusions than the results in the literature.