The Maximum Principle for One Kind of Stochastic Optimization Problem and Application in Dynamic Measure of Risk

The authors get a maximum principle for one kind of stochastic optimization problem motivated by dynamic measure of risk. The dynamic measure of risk to an investor in a financial market can be studied in our framework where the wealth equation may have nonlinear coefficients.     

动态估价的连续性质

在研究风险资产和未定权益的定价问题时,彭实戈[1]提出了动态估价的概念并研究了它的很多性质.在这些结果的基础上,本文进一步研究了动态估价在几乎处处意义下的一些连续性质.     

风险的动态定量和一个相关的随机对策问题

本文讨论不完全市场中股票收益率不确定时的动态风险度量问题和一个相关的随机对策问题。该动态风险度量可表示为一个随机最优控制问题的值函数,以倒向随机微分方程为工具我们给出了最优目标具有的形式,并给出随机对策问题上值与下值相等的充分条件和鞍点的存在性。     

倒向随机微分方程及其应用

本文将介绍一类新的议程：倒向随机微分方程，为了便于理解，我们将首先通过与常微分方程和经典的随机微分方程的对比，并通过数理经济和数学金融学中的一个典型的例子来引入倒向随机微分方程。     

Dynamically consistent nonlinear evaluations with their generating functions in L p

In this paper, we study dynamically consistent nonlinear evaluations in Lp(1p2). One of our aim is to obtain the following result: under a domination condition, an Ft-consistent evaluation is an Eg-evaluation in Lp . Furthermore, without the assumption that the generating function g(t, ω, y, z) is continuous with respect to t, we provide some useful characterizations of an Eg-evaluation by g and give some applications. These results include and extend some existing results.     

Stochastic differential games with reflection and related obstacle problems for Isaacs equations

In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1997), which turns out to be very useful because it allows us to estimate the L ^p -distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.     

伊藤-泊松型随机微分方程的线性二次控制

本文研究伊藤-泊松型随机微分方程的线性二次控制问题,利用动态规划方法、伊藤公式等技巧,通过解HJB方程,我们得到了随机Riccati方程及另外两个微分方程,求出控制变量,解决了线性二次最优控制最优问题.     

Recent Results on Recurrent Solutions and Limit Distributions of SDEs

The limit distribution for homogeneous Markov processes is studied extensively and well understood, but it is not the case for inhomogeneous Markov processes. In this paper, we review some recent results on inhomogeneous Markov processes generated by non-autonomous stochastic (partial) differential equations (SDE in short). Under some suitable conditions, we show that the distribution of recurrent solutions of SDEs constitutes the limit distribution of the corresponding inhomogeneous Markov processes.     

油气田开采系统的随机模型及其求解

本文建立了油气田开采动态系统的随机微分方程数学模型，在对模型求解的基础上，给出了各状态变量的均值函数与方差函数的估计。     

含参数的一类非线性算子方程解对参数的连续相依性定理及应用

该文利用锥理论和Ｂａｎａｃｈ 压缩映象原理讨论了一类含参数的非线性算子方程，证明了这类方程解的存在唯一性及其解对参数的连续相依性定理，并给出了对Ｂａｎａｃｈ 空间中含参数的一阶积－微分方程初值问题的应用．     

Valuation of futures options with initial margin requirements and daily price limit

The paper presents a valuation model of futures options trading at exchanges with initial margin requirements and daily price limit, and this result gives an academic guidance to design trading rules at exchanges. Unlike the leading work of Black, certain trading rules are considered so as to be more fit for practical futures markets. The paper prices futures options with initial margin requirements and daily price limit by duplicating them with the help of the theory of backward stochastic differential equations （BSDEs, for short）. Furthermore, an explicit expression of the price Of the call （or the put） futures option is given and also is shown to be the unique solution of the associated nonlinear partial differential equation.     

A Probabilistic Approach for Nonlinear Equations Involving the Fractional Laplacian and a Singular Operator

We consider a class of nonlinear integro-differential equations involving a fractional power of the Laplacian and a nonlocal quadratic nonlinearity represented by a singular integral operator. Initially, we introduce cut-off versions of this equation, replacing the singular operator by its Lipschitz continuous regularizations. In both cases we show the local existence and global uniqueness in L¹L^p. Then we associate with each regularized equation a stable-process-driven nonlinear diffusion; the law of this nonlinear diffusion has a density which is a global solution in L¹ of the cut-off equation. In the next step we remove the cut-off and show that the above densities converge in a certain space to a solution of the singular equation. In the general case, the result is local, but under a more stringent balance condition relating the dimension, the power of the fractional Laplacian and the degree of the singularity, it is global and gives global existence for the original singular equation. Finally, we associate with the singular equation a nonlinear singular diffusion and prove propagation of chaos to the law of this diffusion for the related cut-off interacting particle systems. Depending on the nature of the singularity in the drift term, we obtain either a strong pathwise result or a weak convergence result. Mathematics Subject Classifications (2000) 60K35, 35S10.     

On optimal solutions of general continuous-singular stochastic control problem of McKean-Vlasov type

In this paper, we establish general necessary optimality conditions for stochastic continuous-singular control of McKean-Vlasov type equations. The coefficients of the state equation depend on the state of the solution process as well as of its probability law and the control variable. The coefficients of the system are nonlinear and depend explicitly on the absolutely continuous component of the control. The control domain under consideration is not assumed to be convex. The proof of our main result is based on the first- and second-order derivatives, with respect to measure in Wasserstein space of probability measures, and by using variational method.     

测度空间上非线性中子迁移方程的适定性和渐近性

本文给出测定空间上非线性中子迁移方程的适定性的充分条件及渐近性质．     

Ergodic BSDE with unbounded and multiplicative underlying diffusion and application to large time behaviour of viscosity solution of HJB equation

We study ergodic backward stochastic differential equations (EBSDEs), for which the underlying diffusion is assumed to be multiplicative and of linear growth. The fact that the forward process has an unbounded diffusion is balanced with an assumption of weak dissipativity for its drift. Moreover, the forward equation is assumed to be non-degenerate. We study the existence and uniqueness of EBSDEs and we apply our results to an ergodic optimal control problem. In particular, we show the large time behaviour of viscosity solution of Hamilton–Jacobi–Bellman equation with an exponential rate of convergence when the underlying diffusion is multiplicative and unbounded.     

Ito 过程在免费邮箱硬盘容量动态分配模型中的应用

分析了目前电子邮箱市场的情况,运用微分方程建立了邮箱用户增长模型,预测某邮箱未来几年内的用户数量及企业需要提供的硬盘容量。从减少成本的角度分析,运用Ito随机过程,对风险进行评估,建立硬盘容量动态分配模型,指导企业如何安排合适的硬盘容量,既能满足市场需要,又达到规避风险和降低成本的目的,提高市场竞争力。     

A Theoretical Model and Experimental Investigation of a Nonlinear Constitutive Equation for Elastic Porous Metal Rubbers

Starting from the theory of porous materials, the nonlinear theory of dry friction, and the model of a bent slender beam, a nonlinear constitutive equation for elastic porous metal rubbers is constructed. Static compressive experiments were carried out on hollow cylinders of such a rubber, from which relations between all coefficients of the constitutive equation and the material density were determined. Based on the data obtained, the constitutive equation of the material was predicted successfully. The effect of density of the metal rubber on its stiffness and on the nonlinear constitutive equation was revealed by experiments.__________Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 4, pp. 449–460, July–August, 2005.     

Risk minimizing portfolios and HJBI equations for stochastic differential games

In this paper, we consider the problem to find a market portfolio that minimizes the convex risk measure of the terminal wealth in a jump diffusion market. We formulate the problem as a two player (zero-sum) stochastic differential game. To help us find a solution, we prove a theorem giving the Hamilton–Jacobi–Bellman–Isaacs (HJBI) conditions for a general zero-sum stochastic differential game in a jump diffusion setting. We then use the theorem to study particular risk minimization problems. Finally, we extend our approach to cover general stochastic differential games (not necessarily zero-sum), and we obtain similar HJBI equations for the Nash equilibria of such games.     

Invariant Measures and Asymptotic Gaussian Bounds for Normal Forms of Stochastic Climate Model

The systematic development of reduced low-dimensional stochastic climate models from observations or comprehensive high dimensional climate models is an important topic for atmospheric low-frequency variability, climate sensitivity, and improved extended range forecasting. Recently, techniques from applied mathematics have been utilized to systematically derive normal forms for reduced stochastic climate models for low-frequency variables. It was shown that dyad and multiplicative triad interactions combine with the climatological linear operator interactions to produce a normal form with both strong nonlinear cubic dissipation and Correlated Additive and Multiplicative (CAM) stochastic noise. The probability distribution functions (PDFs) of low frequency climate variables exhibit small but significant departure from Gaussianity but have asymptotic tails which decay at most like a Gaussian. Here, rigorous upper bounds with Gaussian decay are proved for the invariant measure of general normal form stochastic models. Asymptotic Gaussian lower bounds are also established under suitable hypotheses.