随机能源Logistic反馈控制系统的分岔研究

建立了非线性随机动力模型—带噪声的能源Logistic反馈控制模型,应用随机平均法对随机动力模型进行了简化,得到了一个二维的扩散过程.二维过程满足Ito型随机微分方程,应用不变测度理论研究了该模型的随机分岔.最后,给出了数值实验验证了相应的结论.     

装备空投的一般数学模型及模拟计算

装备投放是一种部队快速生成战斗力的机动方法.投放过程涉及到气动力学、气固耦合、装备属性等领域知识.通过合理简化,对投放过程的四个阶段分别进行受力分析,在二维情况下建立了对应的微分方程组模型,并给出了相关算例.考虑实际风速、风向等随机参数的影响,在三维情况下建立了带参的微分方程模型,并随机模拟了装备降落点的范围.建立的数学模型对定点降落,及时把握空投最佳时机有积极的指导意义.     

基于随机演化博弈的重大工程项目主体行为博弈及仿真分析

从演化博弈视角分析了重大工程项目中利益主体业主与承包商的行为博弈,建立了复制动态方程,对方程引入白噪声来反映演化过程受到的随机干扰,建立了随机动力系统,借鉴Ito随机微分方程来分析博弈双方的策略演化,给出了策略稳定的充分条件,并进行了仿真分析.研究表明:在随机扰动下,当决策主体采取积极风险管理的成本小于补偿成本与分担成本之和时,决策双方的策略会上下波动,最终演化至稳定策略积极风险管理;当决策主体采取积极风险管理的成本大于补偿成本与分担成本之和时,积极风险管理策略不稳定,决策双方会倾向于采取消极风险管理.     

基于个体公平原则的寿险产品定价模型

在无套利框架的基础上,讨论基于个体公平原则下的寿险产品定价问题,即运用倒向随机微分方程理论,将投保人和保险人置于同一系统中进行考虑:首先,根据双方的随机投资决策目标分别建立无套利寿险定价模型和动态资产份额定价模型,得出两个特殊线性倒向随机微分方程的显式解;然后,建立基于个体公平原则的寿险定价模型,从投保人和保险人双方的角度对寿险产品进行公平定价,得出了从供需双方考虑的投资回报定价公式;最后,利用所建立的模型进行案例分析,计算出基于个体公平原则的保费及保险公司的投资策略.该寿险产品定价模型不仅考虑了保险人的意愿,还同时考虑了投保人的实际情况,因此,按此定价理念开发出的保险产品,不仅可以提高产品研发的成功率,而且使得研发出的新产品更能在竞争激烈的保险市场中站稳脚步.     

定价问题和一类倒向随机微分方程解的存在唯一性

本文建立了由一个多维Brown运动、Poisson过程和跳时固定的简单点过程共同驱动的股票价格模型．在此模型下，将未定权益的定价问题归结为一类倒向随机微分方程的求解问题．证明了这类倒向随机微分方程适应解的存在唯一性问题，并给出了一个关于未定权益的定价公式．     

随机利率模型下触发式利率挂钩型理财产品的定价北大核心

主要研究随机利率模型下触发式利率挂钩型理财产品的定价,运用△-对冲技术建立偏微分方程,最终给出随机利率模型下此款理.财产品的定价.     

外汇期权的多维跳-扩散模型

本文建立了外汇期权的多维跳-扩散模型,在此模型下将外汇欧式未定权益的定价问题归结为一类倒向随机微分方程的求解问题,证明了这类倒向随机微分方程适应解的存在唯一性问题,并给出了一个关于外汇欧式未定权益的定价公式.     

随机利率模型下触发式利率挂钩型理财产品的定价

主要研究随机利率模型下触发式利率挂钩型理财产品的定价,运用△-对冲技术建立偏微分方程,最终给出随机利率模型下此款理.财产品的定价.     

由Lvy过程驱动的反射型倒向随机微分方程（英文）

本文给出了由Levy过程驱动的反射型倒向随机微分方程解的存在唯一性,其中反射壁是右连左极且跳跃是任意的.为了证明上述结论,我们建立了由Levy过程驱动的倒向随机微分方程的单调极限定理.     

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.     

A random differential transform method: Theory and applications

In this article, a Differential Transform Method (DTM) based on the mean fourth calculus is developed to solve random differential equations. An analytical mean fourth convergent series solution is found for a nonlinear random Riccati differential equation by using the random DTM. Besides obtaining the series solution of the Riccati equation, we provide approximations of the main statistical functions of the stochastic solution process such as the mean and variance. These approximations are compared to those obtained by the Euler and Monte Carlo methods. It is shown that this method applied to the random Riccati differential equation is more efficient than the two above mentioned methods.     

Ensemble Averaging for Dynamical Systems Under Fast Oscillating Random Boundary Conditions

This article is devoted to providing a theoretical underpinning for ensemble forecasting with rapid fluctuations in body forcing and in boundary conditions. Ensemble averaging principles are proved under suitable “mixing” conditions on random boundary conditions and on random body forcing. The ensemble averaged model is a nonlinear stochastic partial differential equation, with the deviation process (i.e., the approximation error process) quantified as the solution of a linear stochastic partial differential equation.     

Weak approximations. A Malliavin calculus approach

We introduce a variation of the proof for weak approximations that is suitable for studying the densities of stochastic processes which are evaluations of the flow generated by a stochastic differential equation on a random variable that may be anticipating. Our main assumption is that the process and the initial random variable have to be smooth in the Malliavin sense. Furthermore, if the inverse of the Malliavin covariance matrix associated with the process under consideration is sufficiently integrable, then approximations for densities and distributions can also be achieved. We apply these ideas to the case of stochastic differential equations with boundary conditions and the composition of two diffusions.

     

On the Equivalence of Different Approaches to Stochastic Partial Differential Equations

The solution of a stochastic partial differential equation with white noise disturbance can be treated in two different ways: as a real-valued random field or as a function-space valued stochastic process. After introducing these views briefly it is shown that these two approaches are equivalent. Some further results on FOURIER decomposition and on asymptotic behaviour of the linear equation are given and a few comments on the nonlinear case are added.     

Mathematical analysis of the stochastic dynamics of a spinning spherical satellite

The vector differential equation describing the motion of a spinning spherical satellite is here studied by assuming that the aerodynamical forces have random nature. The resulting evolution equation is a random differential equation with stochastic process coefficients which is solved by using a perturbation procedure and by following known methods of stochastic systems analysis. The solution process is therefore found in an approximated analytical form, which allows the determination of some statistical properties of the system.     

A Deterministic Approach To Stochastic Optimal Control With Application To Anticipative Control

Using the decomposition of solution of SDE, we consider the stochastic optimal control problem with anticipative controls as a family of deterministic control problems parametrized by the paths of the driving Wiener process and of a newly introduced Lagrange multiplier stochastic process (nonanticipativity equality constraint). It is shown that the value function of these problems is the unique global solution of a robust equation (random partial differential equation) associated to a linear backward Hamilton-Jacobi-Bellman stochastic partial differential equation (HJB SPDE). This appears as limiting SPDE for a sequence of random HJB PDE's when linear interpolation approximation of the Wiener process is used. Our approach extends the Wong-Zakai type results [20] from SDE to the stochastic dynamic programming equation by showing how this arises as average of the limit of a sequence of deterministic dynamic programming equations. The stochastic characteristics method of Kunita [13] is used to represent the value function. By choosing the Lagrange multiplier equal to its nonanticipative constraint value the usual stochastic (nonanticipative) optimal control and optimal cost are recovered. This suggests a method for solving the anticipative control problems by almost sure deterministic optimal control. We obtain a PDE for the “cost of perfect information” the difference between the cost function of the nonanticipative control problem and the cost of the anticipative problem which satisfies a nonlinear backward HJB SPDE. Poisson bracket conditions are found ensuring this has a global solution. The cost of perfect information is shown to be zero when a Lagrangian submanifold is invariant for the stochastic characteristics. The LQG problem and a nonlinear anticipative control problem are considered as examples in this framework     

Dynamical Analysis for a General Jerky Equation with Random Excitation

A general jerky equation with random excitation is investigated in this paper. Before introducing the random excitation term, the equation is reduced to a two-dimensional model when undergoing a Hopf bifurcation. Then the model with the parametric excitation and external excitation is converted to a stochastic differential equation with singularity based on the stochastic average theory. For the equation, its dynamical behaviors are analyzed in different parameters'' spaces, including the stability, stochastic bifurcation and stationary solution. Besides, numerical simulations are given to show the asymptotic behavior of the stationary solution.     

Parameter and moment bounds for a nonlinear stochastic ecology model

A differential equation model of a marine ecosystem is formulated as a stochastic process. The ecosystem is modeled by considering the random exchange of a chemical nutrient between three components of the ecosystem. The Chapman-Kolmogorov equations and the moment or cumulant generating functions for the process are derived to examine analytically the behavior of the moments of the process. Through the use of differential inequalities, bounds on the exchange rate parameters are derived to reflect component extinction. Bounds on the moments of the process are also obtained.     

Stationary Conjugation of Flows for Parabolic SPDEs with Multiplicative Noise and Some Applications

Abstract

The purpose of this paper is to transform a nonlinear stochastic partial differential equation of parabolic type with multiplicative noise into a random partial differential equation by using a bijective random process. A stationary conjugation is constructed, which is of interest for asymptotic problems. The conjugation is used here to prove the existence of the stochastic flow, the perfect cocycle property and the existence of the random attractor, all nontrivial properties in the case of multiplicative noise.     

APPROXIMATION THEOREMS BASED ON RANDOM PARTITIONS FOR STOCHASTIC DIFFERENTIAL EQUATION AND THEIR APPLICATIONS

This kind of problems is discussed:When we use certain smooth approximations of theBrownian motion W as substitutes for it in stochastic line integral and stochastic differentialequation,do these resultant integrals and solutions converge to the original one?Thecorresponding approximation theoroms for two kinds of apprximations are proved,whichare wider than those discussed in[1].Some limit theorems about stochastic line integral andsolutions of stochastic differential equations with respect to random walks are obtained byusing the idea of“embeding a random walk into the Brownian motion”first proposed byA.V.Skorohod.It seems to be remarkable that the method used here is not only effectivefor the one dimensional case,but also for the multi-dimensional case.