Externality and Hamilton-Jacobi equations

The relationship between optimal control problems and Hamilton-Jacobi-Bellman equations is well known [9]. In fact the value function, defined as the infimum of the cost functional, satisfies in the viscosity sense an appropriate Hamilton-Jacobi-Bellman equation. In this paper we consider several control problems such that the cost functional associated to each problem depends explicitly on the value functions of the other problems. This leads to a system of Hamilton-Jacobi-Bellman equations. This is known, in economic context [14] cap XI, as an externality problem. In these problems may occur a lack of uniqueness of the value functions. We give conditions to ensure existence, uniqueness of the value functions and an implicit integral representation formula. Moreover, under uniqueness assumption, we prove that the variational solutions of the associated Hamilton-Jacobi system converge asymptotically to the value functions. We prove also an uniqueness theorem in the case of viscosity solutions of Hamilton-Jacobi-Bellman system.     

Global optimization of arborescent multilevel inventory systems

We consider the numerical resolution of hierarchical inventory problems under global optimization. First we describe the model as well as the dynamical stochastic system and the impulse controls involved. Next we characterize the optimal cost function and we formulate the Hamilton-Jacobi-Bellman equations. We present a numerical scheme and a fast algorithm of resolution, with a result on the speed of convergence. Finally, we apply the discretization method to some examples where we show the usefulness of the proposed numerical method as well as the advantages of operating under global optimization.     

Numerical solution of a minimax ergodic optimal control problem

In this work we consider an L^∞ minimax ergodic optimal control problem with cumulative cost. We approximate the cost function as a limit of evolutions problems. We present the associated Hamilton-Jacobi-Bellman equation and we prove that it has a unique solution in the viscosity sense. As this HJB equation is consistent with a numerical procedure, we use this discretization to obtain a procedure for the primitive problem. For the numerical solution of the ergodic version we need a perturbation of the instantaneous cost function. We give an appropriate selection of the discretization and penalization parameters to obtain discrete solutions that converge to the optimal cost. We present numerical results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic programming in stochastic control of systems with delay

We consider optimal control problems for systems described by stochastic differential equations with delay (SDDE). We prove a version of Bellman's principle of optimality (the dynamic programming principle) for a general class of such problems. That the class in general means that both the dynamics and the cost depends on the past in a general way. As an application, we study systems where the value function depends on the past only through some weighted average. For such systems we obtain a Hamilton-Jacobi-Bellman partial differential equation that the value function must solve if it is smooth enough. The weak uniqueness of the SDDEs we consider is our main tool in proving the result. Notions of strong and weak uniqueness for SDDEs are introduced, and we prove that strong uniqueness implies weak uniqueness, just as for ordinary stochastic differential equations.     

Optimal Control for Utility Portfolio Selection with Liability

In this paper we use stochastic optimal control theory to investigate a dynamic portfolio selection problem with liability process, in which the liability process is assumed to be a geometric Brownian motion and completely correlated with stock prices. We apply dynamic programming principle to obtain Hamilton-Jacobi-Bellman (HJB) equations for the value function and systematically study the optimal investment strategies for power utility, exponential utility and logarithm utility. Firstly, the explicit expressions of the optimal portfolios for power utility and exponential utility are obtained by applying variable change technique to solve corresponding HJB equations. Secondly, we apply Legendre transform and dual approach to derive the optimal portfolio for logarithm utility. Finally, numerical examples are given to illustrate the results obtained and analyze the effects of the market parameters on the optimal portfolios.     

Alternating direction algorithms for solving Hamilton-Jacobi-Bellman equations

We focus on numerically solving a typical type of Hamilton-Jacobi-Bellman (HJB) equations arising from a class of optimal controls with a standard multidimensional diffusion model. Solving such an equation results in the value function and an optimal feedback control law. The Bellman's curse of dimensionality seems to be the main obstacle to applicability of most numerical algorithms for solving HJB. We decompose HJB into a number of lower-dimensional problems, and discuss how the usual alternating direction method can be extended for solving HJB. We present some convergence results, as well as preliminary experimental outcomes.This research was funded in part by an RGC grant from the University of Alabama.     

Dividend Maximization when Cash Reserves Follow a Jump-diffusion Process

This paper deals with the dividend optimization problem for an insurance company, whose surplus follows a jump-diffusion process. The objective of the company is to maximize the expected total discounted dividends paid out until the time of ruin. Under concavity assumption on the optimal value function, the paper states some general properties and, in particular, smoothness results on the optimal value function, whose analysis mainly relies on viscosity solutions of the associated Hamilton-Jacobi-Bellman （HJB） equations. Based on these properties, the explicit expression of the optimal value function is obtained. And some numerical calculations are presented as the application of the results.     

Large deviations for stochastic PDE with Lévy noise

We prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small Lévy noise. We use general large deviations theorems of Varadhan and Bryc coupled with the techniques of Feng and Kurtz (2006) [15], viscosity solutions of integro-partial differential equations in Hilbert spaces, and deterministic optimal control methods. The Laplace limit is identified as a viscosity solution of a Hamilton-Jacobi-Bellman equation of an associated control problem. We also establish exponential moment estimates for solutions of stochastic evolution equations driven by Lévy noise. General results are applied to stochastic hyperbolic equations perturbed by subordinated Wiener process.     

Optimal portfolio and consumption selection with default risk

We investigate an optimal portfolio and consumption choice problem with a defaultable security. Under the goal of maximizing the expected discounted utility of the average past consumption, a dynamic programming principle is applied to derive a pair of second-order parabolic Hamilton-Jacobi-Bellman (HJB) equations with gradient constraints. We explore these HJB equations by a viscosity solution approach and characterize the post-default and pre-default value functions as a unique pair of constrained viscosity solutions to the HJB equations.     

带有交易成本和红利的最优消费投资策略研究

本文利用HJB方程粘性解理论,考虑带有红利收益和交易成本后,对现有最优消费投资模型作了推广,研究了投资者在带有红利和交易成本情形下的最优消费投资策略。     

Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process

In this paper, we study the optimal investment and proportional reinsurance strategy when an insurance company wishes to maximize the expected exponential utility of the terminal wealth. It is assumed that the instantaneous rate of investment return follows an Ornstein-Uhlenbeck process. Using stochastic control theory and Hamilton-Jacobi-Bellman equations, explicit expressions for the optimal strategy and value function are derived not only for the compound Poisson risk model but also for the Brownian motion risk model. Further, we investigate the partially observable optimization problem, and also obtain explicit expressions for the optimal results.     

Multi-asset investment-consumption model with transaction costs

In this paper, we consider the multi-asset optimal investment-consumption model: a riskless asset and d risky assets. when the initial time is t?0, for a proportional transaction costs and discount factors, we proof that the value function of the model is a unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equations.     

Optimal dynamic pricing of inventories with stochastic demand and discounted criterion

We consider a continuous time dynamic pricing problem for selling a given number of items over a finite or infinite time horizon. The demand is price sensitive and follows a non-homogeneous Poisson process. We formulate this problem as to maximize the expected discounted revenue and obtain the structural properties of the optimal revenue function and optimal price policy by the Hamilton-Jacobi-Bellman (HJB) equation. Moreover, we study the impact of the discount rate on the optimal revenue function and the optimal price. Further, we extend the problem to the case with discounting and time-varying demand, the infinite time horizon problem. Numerical examples are used to illustrate our analytical results.     

Uniqueness of solution of production control problem in a manufacturing system with degenerate demand

This paper studies the production inventory problem of minimizing the expected discounted present value of production cost control in a manufacturing system with degenerate stochastic demand. We establish the existence of a unique solution of the Hamilton-Jacobi-Bellman (HJB) equation associated with this problem. The optimal control is given by a solution to the corresponding HJB equation.     

Manage Pension Deficit with Heterogeneous Insurance

This paper considers a positive and increasing pension deficit of a certain pay-as-you-go (PAYG) pension system, and tries to make up for this deficit by using heterogeneous insurance. The positive pension deficit is formulated as a mathematical function in continuous time. The surplus of an appropriate heterogeneous insurance is described by diffusion approximation of a Cramér-Lundberg process. The system of extended Hamilton-Jacobi-Bellman equations under mean-variance criterion is established. The closed-form solution and optimal surplus-multiplier of heterogenous insurance are obtained. Some interpretations further explain the theoretical values of the results.

     

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems is proposed based on the extended Hamiltonian principle, the Hamilton-Jacobi-Bellman (HJB) equation and its variational integral equation, and the finite time element approximation. The differential extended Hamiltonian equations for structural vibration systems are replaced by the variational integral equation, which can preserve intrinsic system structure. The optimal control law dependent on the value function is determined by the HJB equation so as to satisfy the overall optimality principle. The partial differential equation for the value function is converted into the integral equation with variational weighting. Then the successive solution of optimal control with system state is designed. The two variational integral equations are applied to sequential time elements and transformed into the algebraic equations by using the finite time element approximation. The direct optimal control on each time element is obtained respectively by solving the algebraic equations, which is unconstrained by the system state observed. The proposed control algorithm is applicable to linear and nonlinear systems with the quadratic performance index, and takes into account the effects of external excitations measured on control. Numerical examples are given to illustrate the optimal control effectiveness.     

Hamilton-Jacobi-Bellman方程的区域分解算法

对离散Hamilton-Jacobi-Bellman方程提出了一类区域分解算法,并在合理的假设下证明了该算法的单调收敛性,数值结果表明该算法的有效性与准确性．     

The Optimal Dividend and Capital Injection Strategies in the Classical Risk Model with Randomized Observation Periods

This paper considers the optimal dividend and capital injection strategies in the classical risk model with randomized observation periods. Assume that ruin is prohibited. We aim to maximise the expected discounted dividend payments minus the expected penalised discounted capital injections. We derive the associated Hamilton-Jacobi-Bellman (HJB) equation and prove the verification theorem. The optimal control strategy and the optimal value function are obtained under the assumption that the claim sizes are exponentially distributed.     

带利率和随机观测时间的布朗运动模型中最优分红策略

本文研究了带利率和随机观测时间的布朗运动模型中的最优分红问题.利用随机控制理论,获得了最优值函数相应的HJB方程,表明最优分红策略是障碍策略,并给出了最优值函数的显式表达式,推广了文献[19]的结果.