A diffusion approximation result for two parameter processes

Summary We consider a one-dimensional linear wave equation with a small mean zero dissipative field and with the boundary condition imposed by the so-called Goursat problem. In order to observe the effect of the randomness on the solution we perform a space-time rescaling and we rewrite the problem in a diffusion approximation form for two parameter processes. We prove that the solution converges in distribution toward the solution of a two-parameter stochastic differential equation which we identify. The diffusion approximation results for oneparameter processes are well known and well understood. In fact, the solution of the one-parameter analog of the problem we consider here is immediate. Unfortunately, the situation is much more complicated for two-parameter processes and we believe that our result is the first one of its kind.Partially supported by ONR N00014-91-J-1010     

The maximum principle for a jump-diffusion mean-field model and its application to the mean–variance problem

This paper establishes a necessary and sufficient stochastic maximum principle for a mean-field model with randomness described by Brownian motions and Poisson jumps. We also prove the existence and uniqueness of the solution to a jump-diffusion mean-field backward stochastic differential equation. A new version of the sufficient stochastic maximum principle, which only requires the terminal cost is convex in an expected sense, is applied to solve a bicriteria mean–variance portfolio selection problem.     

Lower bound for the mean project completion time in dynamic PERT networks

We apply the stochastic dynamic programming to obtain a lower bound for the mean project completion time in a PERT network, where the activity durations are exponentially distributed random variables. Moreover, these random variables are non-static in that the distributions themselves vary according to some randomness in society like strike or inflation. This social randomness is modelled as a function of a separate continuous-time Markov process over the time horizon. The results are verified by simulation.     

Large deviation principles for the stochastic quasi-geostrophic equations

In this paper we establish the large deviation principle for the stochastic quasi-geostrophic equation with small multiplicative noise in the subcritical case. The proof is mainly based on the weak convergence approach. Some analogous results are also obtained for the small time asymptotics of the stochastic quasi-geostrophic equation.     

Stochastic evolution of innovation diffusion in heterogeneous groups: study of life cycle patterns

A theoretical framework has been proposed to study patternsof innovation diffusion in a heterogeneous population, withapplicability to a number of problem areas including marketing.The heterogeneity in the population is captured through randomlyvarying parameters, which have been modelled in terms of two-pointdistributions. The effect of heterogeneity leads to the generationof bi-modal life cycle patterns besides the conventional uni-modalpattern resulting from S-shaped curve. The stochastic evolutionof the mean and variance of the number of adopters is foundto depict a high level of relative fluctuation around the pointof inflexion. As a result of randomness in parameters, the resultingdifferential equation for the evolution of the mean of the adoptionprocess is characterized by a non-autonomous system having parameterswhich are no longer constant but become time dependent. Fordemonstrating the effectiveness of the proposed framework, areal data set which depicts a bi-modal life cycle curve is investigated.The fit is found to be extremely good while capturing appropriateproduct life cycle curve.     

A computational analysis for mean exit time under non-Gaussian Lévy noises

Complex dynamical systems are often subject to non-Gaussian random fluctuations. The exit phenomenon, i.e., escaping from a bounded domain in state space, is an impact of randomness on the evolution of these dynamical systems. The existing work is about asymptotic estimate on mean exit time when the noise intensity is sufficiently small. In the present paper, however, the authors analyze mean exit time for arbitrary noise intensity, via numerical investigation. The mean exit time for a dynamical system, driven by a non-Gaussian, discontinuous (with jumps), α-stable Lévy motion, is described by a differential equation with nonlocal interactions. A numerical approach for solving this nonlocal problem is proposed. A computational analysis is conducted to investigate the relative importance of jump measure, diffusion coefficient and non-Gaussianity in affecting mean exit time.     

连续SV模型下衍生证券定价

1973年 Black- Scholes公式的出现极大推动衍生证券的发展 ,该公式的不足是假设影响标的资产价格波动的扩散系数为常数 ;80年代后期的 SV模型是针对该问题的离散统计模型。本文在两者的基础上讨论SV模型和 Black- Scholes公式结合。在讨论一般化衍生证券定价的基础上 ,通过 SV模型的连续化 ,构造一个2维随机微分方程 ,最后讨论了一种可以接受的数值计算方法     

带乘性噪声的空间分数阶随机非线性Schrödinger方程的广义多辛算法

带乘性噪声的空间分数阶随机非线性Schrödinger方程是一类重要的方程,可应用于描述开放非局部量子系统的演化过程.该方程为一个无穷维分数阶随机Hamilton系统,且具有广义多辛结构和质量守恒的性质.针对该方程的广义多辛形式,在空间上采用拟谱方法离散分数阶微分算子,在时间上则采用隐式中点格式,构造出一类保持全局质量的广义多辛格式.对行波解和平面波解等进行数值模拟,结果验证了所构造格式的有效性和保结构性质,时间均方收敛阶约在0.5到1之间.     

One-dimensional stochastic heat equation with discontinuous conductance

We introduce a new stochastic partial differential equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space–time white noise. Such equation could be used in mathematical modeling of diffusion phenomena in medium consisting of two kinds of materials and undergoing stochastic perturbations. We prove the existence of the solution and we present explicit expressions of its covariance and variance functions. Some regularity properties of the solution sample paths are also analyzed.     

Mean–variance portfolio selection under a constant elasticity of variance model

This paper discusses a mean–variance portfolio selection problem under a constant elasticity of variance model. A backward stochastic Riccati equation is first considered. Then we relate the solution of the associated stochastic control problem to that of the backward stochastic Riccati equation. Finally, explicit expressions of the optimal portfolio strategy, the value function and the efficient frontier of the mean–variance problem are expressed in terms of the solution of the backward stochastic Riccati equation.     

Stochastic Schrodinger Equation for a Quantum Oscillator with Dissipation

In this paper, we construct an exact solution of the stochastic Schrodinger equation for a quantum oscillator with possible dissipation of energy taken into account. Using the explicit form of the solution, we calculate estimates for the characteristic damping time of free damped oscillations. In the case of forced oscillations, we obtain formulas for the Q-factor of the system and for the variance of the coordinate and momentum of a quantum oscillator with dissipation. We obtain the quantum analog of the classical diffusion equation and explicitly show that the equations of motion for the mean value of the momentum operator following from the solution of the stochastic Schrodinger equation play the role of the quantum Langevin equation describing Brownian motion under the action of a stochastic force.     

What Is a Quantum Stochastic Differential Equation from the Point of View of Functional Analysis?

We prove that a quantum stochastic differential equation is the interaction representation of the Cauchy problem for the Schrödinger equation with Hamiltonian given by a certain operator restricted by a boundary condition. If the deficiency index of the boundary-value problem is trivial, then the corresponding quantum stochastic differential equation has a unique unitary solution. Therefore, by the deficiency index of a quantum stochastic differential equation we mean the deficiency index of the related symmetric boundary-value problem.In this paper, conditions sufficient for the essential self-adjointness of the symmetric boundary-value problem are obtained. These conditions are closely related to nonexplosion conditions for the pair of master Markov equations that we canonically assign to the quantum stochastic differential equation.     

Bogoliubov averaging principle of stochastic reaction–diffusion equation

The purpose of this paper is to establish Bogoliubov averaging principle of stochastic reaction–diffusion equation with a stochastic process and a small parameter. The solutions to stochastic reaction–diffusion equation can be approximated by solutions to averaged stochastic reaction–diffusion equation in the sense of convergence in probability and in distribution. Namely, we establish a weak law of large numbers for the solution of stochastic reaction–diffusion equation.     

A Central Limit Theorem for Stochastic Heat Equations in Random Environment

In this article, we investigate the asymptotic behavior of the solution to a one-dimensional stochastic heat equation with random nonlinear term generated by a stationary, ergodic random field. We extend the well-known central limit theorem for finite-dimensional diffusions in random environment to this infinite-dimensional setting. Due to our result, a central limit theorem in \(L^1\) sense with respect to the randomness of the environment holds under a diffusive time scaling. The limit distribution is a centered Gaussian law whose covariance operator is explicitly described. The distribution concentrates only on the space of constant functions.     

随机时滞控制系统的均方BIBO稳定性

本文研究了具有时滞和非线性扰动的随机控制系统的均方有界输入-有界输出(BIBO)稳定.首先,探讨了具有离散时滞和非线性扰动的随机系统的均方BIBO稳定性问题,在此基础上,进一步研究带有离散时滞和分布时滞以及非线性扰动的随机系统的均方BIBO稳定性.通过设计合理的控制器,建立合适的Lyapunov泛函,结合Riccati矩阵方程,得到时滞依赖的均方BIBO稳定性条件.     

一类随机利率下的破产时罚金折现期望

本文在经典风险模型下, 引进带有一种随机利率的破产时罚金折现期望的概念, 其利率的随机性通过标准Wiener过程和Poisson过程来描述. 给出破产时罚金折现期望所满足的更新方程, 并利用这个更新方程给出破产时罚金折现期望的渐近公式.     

Lévy-driven Volterra Equations in Space and Time

We investigate nonlinear stochastic Volterra equations in space and time that are driven by Lévy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend on integrability properties of the kernel and the characteristics of the Lévy basis. Particular attention is devoted to equations with stationary solutions, or more generally, to equations with infinite memory, that is, where the time domain of integration starts at minus infinity. Here, in contrast to the case where time is positive, the usual integrability conditions on the kernel are no longer sufficient for the existence and uniqueness of solutions, but we have to impose additional size conditions on the kernel and the Lévy characteristics. Furthermore, once the existence of a solution is guaranteed, we analyze its asymptotic stability, that is, whether its moments remain bounded when time goes to infinity. Stability is proved whenever kernel and characteristics are small enough, or the nonlinearity of the equation exhibits a fractional growth of order strictly smaller than one. The results are applied to the stochastic heat equation for illustration.     

Stochastic optimal control of first-passage failure for coupled Duffing–van der Pol system under Gaussian white noise excitations

In this paper, the stochastic averaging method of quasi-non-integrable-Hamiltonian systems is applied to Duffing–van der Pol system to obtain partially averaged Ito stochastic differential equations. On the basis of the stochastic dynamical programming principle and the partially averaged Ito equation, dynamical programming equations for the reliability function and the mean first-passage time of controlled system are established. Then a non-linear stochastic optimal control strategy for coupled Duffing–van der Pol system subject to Gaussian white noise excitation is taken for investigating feedback minimization of first-passage failure. By averaging the terms involving control forces and replacing control forces by the optimal ones, the fully averaged Ito equation is derived. Thus, the feedback minimization for first-passage failure of controlled system can be obtained by solving the final dynamical programming equations. Numerical results for first-passage reliability function and mean first-passage time of the controlled and uncontrolled systems are compared in illustrative figures to show effectiveness and efficiency of the proposed method.     

利率均值回复模型下标的资产期权定价

在利率均值回复金融市场中 ,给出了财富贴现过程的随机微分方程 ;证明了与之联系的倒向随机微分方程解的存在唯一性 .最后 ,从倒向随机微分方程的解出发 ,得到了欧式期权定价的条件期望定价公式 .