STABILITY IN LAGRANGE SENSE FOR A CLASS OF STOCHASTIC STATIC NEURAL NETWORKS WITH MIXED TIME DELAYS

In this paper,the stability in Lagrange sense of a class of stochastic static neural networks with mixed time delays is studied.Based on the Lyapunov stability theory and with the help of stochastic analysis technique,the criteria for the stability in Lagrange sense of stochastic static neural networks with mixed time delays is obtained.One example is given to verify the advantage and applicability of the proposed results.     

Lagrange方法和期权定价

本文对于连续时间的随机最优控制问题，建立起随机Lagrange方法。然后，利用该方法讨论了欧洲期权的定价问题。     

Asset allocation for a DC pension fund with stochastic income and mortality risk: A multi-period mean–variance framework

This paper investigates an asset allocation problem for defined contribution pension funds with stochastic income and mortality risk under a multi-period mean–variance framework. Different from most studies in the literature where the expected utility is maximized or the risk measured by the quadratic mean deviation is minimized, we consider synthetically both to enhance the return and to control the risk by the mean–variance criterion. First, we obtain the analytical expressions for the efficient investment strategy and the efficient frontier by adopting the Lagrange dual theory, the state variable transformation technique and the stochastic optimal control method. Then, we discuss some special cases under our model. Finally, a numerical example is presented to illustrate the results obtained in this paper.     

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     

LAGRANGE STABILITY IN MEAN SQUARE OF STOCHASTIC REACTION DIFFUSION EQUATIONS

This work is devoted to the discussion of stochastic reaction diffusion equations and some new theorems on Lagrange stability in mean square of the solution are established via Lyapunov method which is nothing to be done in the past.     

基于动态VaR约束与随机波动率模型的最优投资策略

研究Stein-Stein随机波动率模型下带动态VaR约束的最优投资组合选择问题. 假设投资者的目标是最大化终端财富的期望幂效用,可投资于无风险资产和一种风险资产, 风险资产的价格过程由Stein-Stein随机波动率模型刻画. 同时, 投资者期望能在投资过程中利用动态VaR约束控制所面对的风险.运用Bellman动态规划方法和Lagrange乘子法, 得到了该约束问题最优策略的解析式及特殊情形下最优值函数的解析式; 并通过理论分析和数值算例, 阐述了动态VaR约束与随机波动率对最优投资策略的影响.     

Extended Entropy Functional for Nonlinear Systems in Stochastic Dynamics

The maximum entropy approach is utilized for deriving the stationary probability density function of nonlinear stochastic systems to white noise excitation. To this aim a variational formulation is proposed where by means of the Lagrange multiplier methods the entropy functional is maximised constrained to the Fokker Planck equation. Some exact solutions in terms of Lagrange function of MDOF linear systems and for a class of SDOF nonlinear systems, are obtained.     

具有随机收入的扰动更新风险模型(英文)

本文研究了一类随机收入的扰动更新风险模型的破产问题.运用拉普拉斯变换以及拉格朗日差值公式得到了Gerber-Shiu函数的拉普拉斯变换的渐近表达式,推广了文献[4]中的结论.     

Approximate Lagrange multiplier algorithm for stochastic programs with complete recourse: Nonlinear deterministic constraints

In this paper we design an approximation method for solving stochastic programs with complete recourse and nonlinear deterministic constraints. This method is obtained by combining approximation method and Lagrange multiplier algorithm of Bertsekas type. Thus this method has the advantages of both the two.This project is supported by the National Natural Science Foundation of China.     

On generalized Nash equilibrium in infinite dimension: the Lagrange multipliers approach

In this note we study a class of generalized Nash equilibrium problems and characterize the solutions which have the property that all players share the same Lagrange multipliers. Nash equilibria of this kind were introduced by Rosen in 1965, in finite-dimensional spaces. In order to obtain the same property in infinite dimension, we use very recent developments of a new duality theory. In view of its usefulness in the study of time-dependent or stochastic equilibrium problems, an application in Lebesgue spaces is given.     

Optimization with a class of multivariate integral stochastic order constraints

We study convex optimization problems with a class of multivariate integral stochastic order constraints defined in terms of parametrized families of increasing concave functions. We show that utility functions act as the Lagrange multipliers of the stochastic order constraints in this general setting, and that the dual problem is a search over utility functions. Practical implementation issues are discussed.     

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial chaos order, in any subdomain which does not contain the random boundaries.     

随机波动率市场存在股票误价时的最优投资策略

本文研究了随机波动率市场中存在股票误价(mispricing)时的最优投资组合选择问题.假设投资者的目标是最大化终端财富的期望幂效用;其可投资于无风险资产、市场指数和两支相同权益或近似度极高的股票,其中至少有一支股票存在误价;市场收益的波动率和股票系统风险由Heston随机波动率模型刻画.运用动态规划方法和Lagrange乘子法,分别得到不存在/存在有限卖空约束时,投资者的最优投资策略及最优值函数的解析式,并通过理论分析和数值算例,阐述了投资时间水平和价格随机误差对最优投资策略的影响.     

Plongement stochastique des systèmes lagrangiens

We define an operator which extends classical differentiation from smooth deterministic functions to certain stochastic processes. Based on this operator, we define a procedure which associates a stochastic analog to standard differential operators and ordinary differential equations. We call this procedure stochastic embedding. By embedding Lagrangian systems, we obtain a stochastic Euler–Lagrange equation which, in the case of natural Lagrangian systems, is called the embedded Newton equation. This equation contains the stochastic Newton equation introduced by Nelson in his dynamical theory of Brownian diffusions. Finally, we consider a diffusion with a gradient drift, a constant diffusion coefficient and having a probability density function. We prove that a necessary condition for this diffusion to solve the embedded Newton equation is that its density be the square of the modulus of a wave function solution of a linear Schrödinger equation. To cite this article: J. Cresson, S. Darses, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

A self-dual variational approach to stochastic partial differential equations

Unlike many of their deterministic counterparts, stochastic partial differential equations are not amenable to the methods of calculus of variations à la Euler–Lagrange. In this paper, we show how self-dual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields. We construct solutions as minima of suitable non-negative and self-dual energy functionals on Itô spaces of stochastic processes. We show how a stochastic version of Bolza's duality leads to solutions for equations with additive noise. We then use a Hamiltonian formulation to construct solutions for non-linear equations with non-additive noise such as the stochastic Navier–Stokes equations in dimension two.     

Mean-Variance Portfolio Selection with a Stochastic Cash Flow in a Markov-switching Jump–Diffusion Market

This paper considers a non-self-financing mean-variance portfolio selection problem in which the stock price and the stochastic cash flow follow a Markov-modulated Lévy process and a Markov-modulated Brownian motion with drift, respectively. The stochastic cash flow can be explained as the stochastic income or liability of the investors during the investment process. The existence of optimal solutions is analyzed, and the optimal strategy and the efficient frontier are derived in closed-form by the Lagrange multiplier technique and the LQ (Linear Quadratic) technique.     

Inverse stochastic dominance constraints and rank dependent expected utility theory

We consider optimization problems with second order stochastic dominance constraints formulated as a relation of Lorenz curves. We characterize the relation in terms of rank dependent utility functions, which generalize Yaari's utility functions. We develop optimality conditions and duality theory for problems with Lorenz dominance constraints. We prove that Lagrange multipliers associated with these constraints can be identified with rank dependent utility functions. The problem is numerically tractable in the case of discrete distributions with equally probable realizations. Research supported by the NSF awards DMS-0303545, DMS-0303728, DMI-0354500 and DMI-0354678.     

一个求解条件极值问题的极值点的新方法

利用齐次线性方程组理论,建立了一个求解条件极值问题的极值点的新方法.该方法的优点是:能有效地避免在运用Lagrange乘数法求解条件极值时,因引进了参数而给解方程组带来的困扰.也可以说,对于有些问题我们仅从已知条件入手,不必引进参数就可以直接求得极值点.     

Optimization with multivariate stochastic dominance constraints

We consider stochastic optimization problems where risk-aversion is expressed by a stochastic ordering constraint. The constraint requires that a random vector depending on our decisions stochastically dominates a given benchmark random vector. We identify a suitable multivariate stochastic order and describe its generator in terms of von Neumann–Morgenstern utility functions. We develop necessary and sufficient conditions of optimality and duality relations for optimization problems with this constraint. Assuming convexity we show that the Lagrange multipliers corresponding to dominance constraints are elements of the generator of this order, thus refining and generalizing earlier results for optimization under univariate stochastic dominance constraints. Furthermore, we obtain necessary conditions of optimality for non-convex problems under additional smoothness assumptions.