Optimal consumption of the finite time horizon Ramsey problem

In this paper, we study the stochastic Ramsey problem related to an economic growth model with the CES production function in a finite time horizon. By changing variables, the Hamilton-Jacobi-Bellman equation associated with this optimization problem is transformed. By the viscosity solution technique, we show the existence of a classical solution of the transformed Hamilton-Jacobi-Bellman equation, and then give an optimal consumption policy of the original problem.     

Optimal investment and consumption models with non-linear stock dynamics

We study a generalization of the Merton's original problem of optimal consumption and portfolio choice for a single investor in an intertemporal economy. The agent trades between a bond and a stock account and he may consume out of his bond holdings. The price of the bond is deterministic as opposed to the stock price which is modelled as a diffusion process. The main assumption is that the coefficients of the stock price diffusion are arbitrary nonlinear functions of the underlying process. The investor's goal is to maximize his expected utility from terminal wealth and/or his expected utility of intermediate consumption. The individual preferences are of Constant Relative Risk Aversion (CRRA) type for both the consumption stream and the terminal wealth. Employing a novel transformation, we are able to produce closed form solutions for the value function and the optimal policies. In the absence of intermediate consumption, the value function can be expressed in terms of a power of the solution of a homogeneous linear parabolic equation. When intermediate consumption is allowed, the value function is expressed via the solution of a non-homogeneous linear parabolic equation.     

Optimal economic growth with foreign trade and a learning industry

Ryder (Ref. 1) has developed a simple two-sector macroeconomic model incorporatinglearning by doing as a determinant of the growth of productivity-enhancing experience and its effect on foreign trade. In this paper, optimal foreign trade and capital allocation policies are determined, in the context of the above model, for ranges of the international trade price not considered by Ryder. An extension of Ryder's model to include a dual trade price structure is briefly discussed. A specific numerical example is used to ascertain the configuration of the various features occurring in the extremal fields at different price levels.     

A strong law of large numbers for arrays of rowwise negatively dependent random variables

A strong law of large numbers for arrays of rowwise negatively dependent random variables is obtained which relaxes the usual assumption of rowwise independence. The moment conditions of the main result are similar to previous results, and the stochastic bounded condition also provides a relaxation of the usual distributional assumptions.     

Minimal-time function in problems without local controllability

We consider a linear optimal-time problemP ₀, with control constraint setU ₀ which is not a neighborhood of the origin. We show that it is possible to findT ₀, the minimal-time function or Bellman's function of problemP ₀, as limit in ^m of {T _n } _n1, the sequence of Bellman's functions of linear optimal-time problemsP _n . ProblemsP _n , with perturbed constraint setU _n , satisfy the assumptions givingT _n as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation of problemsP _n .     

Some Remark on Optimal Stochastic Control with Partial Information

Abstract

We are interested in the control problem of a partially observable diffusion process, which is initialized at a fixed point of ? ⁿ , and we want to characterize the associated value function. To resort to the theory of viscosity solutions depends on the possibility to translate such a problem on Hilbert spaces like L ²(? ⁿ ), and so it can not be used here. Nevertheless, a result of N. Bouleau and F. Hirsch allows us to introduce a broadened problem which fulfills the condition. The fact remains to link these two control problems.     

Uniform convergence for the incompressible limit of a tumor growth model

We study a model introduced by Perthame and Vauchelet [19] that describes the growth of a tumor governed by Brinkman's Law, which takes into account friction between the tumor cells. We adopt the viscosity solution approach to establish an optimal uniform convergence result of the tumor density as well as the pressure in the incompressible limit. The system lacks standard maximum principle, and thus modification of the usual approach is necessary.     

On Aronsson Equation and Deterministic Optimal Control

When Hamiltonians are nonsmooth, we define viscosity solutions of the Aronsson equation and prove that value functions of the corresponding deterministic optimal control problems are solutions if they are bilateral viscosity solutions of the Hamilton-Jacobi-Bellman equation. We characterize such a property in several ways, in particular it follows that a value function which is an absolute minimizer is a bilateral viscosity solution of the HJB equation and these two properties are often equivalent. We also determine that bilateral solutions of HJB equations are unique among absolute minimizers with prescribed boundary conditions. This research was partially supported by MIUR-Prin project “Metodi di viscosità, metrici e di teoria del controllo in equazioni alle derivate parziali nonlineari”.     

An application of traveling wave analysis in economic growth model

A temporal–spatial economic growth model is established in this paper. As a useful tool, traveling wave analysis is used to analyze technological growth and diffusion. Numerical simulation shows that this model has perfect performance.     

Nonturnpike Optimal Solutions and Their Approximations in Infinite Horizon

Recently, the authors have proposed a new necessary and sufficient condition for turnpike optimality in calculus of variations with singular Euler equation. The method is based on a characterization of the value function and generalizes the well known method based on the Green theorem. Furthermore, it allows the optimality of a competition between several turnpikes to be characterized. For a class of such problems not enjoying the turnpike property, we give an explicit formula for the value function and show how to characterize the optimal solution as the limiting solution of a family of perturbed problems satisfying the turnpike property. The considered problems are scalar with infinite horizon.     

跳扩散盈余过程的最优投资和最优再保险

站在保险人的立场上,研究了跳扩散盈余过程的最优投资和最优再保险问题.在方差保费原理下,以盈余终值的期望指数效用达到最大作为最优准则,给出了最优策略和值函数的近似表达式.同时也证明了投资总比不投资好的结论.最后,通过一些数例和图表来进一步说明所获得的结论.     

On consumption/investment problems with long-term time-average utilities

We study consumption/investment problems with long-term time-average utilities. The associated Hamilton-Jacobi-Bellman equation can be solved under some regularity conditions of utility rate function, and the optimal portfolio and consumption-rates are exhibited in explicit forms. An application to the optimization problem with finite horizon is also given     

A New Proof for the Equivalence of Weak and Viscosity Solutions for the p-Laplace Equation

In this paper, we give a new proof for the fact that the distributional weak solutions and the viscosity solutions of the p-Laplace equation ?div(|Du| ^p?2 Du) = 0 coincide. Our proof is more direct and transparent than the original proof of Juutinen et al. [8 Juutinen , P. , Lindqvist , P. , Manfredi , J.J. ( 2001 ). On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation . SIAM J. Math. Anal. 33 : 699 – 717 .[Crossref], [Web of Science ®] , [Google Scholar]], which relied on the full uniqueness machinery of the theory of viscosity solutions. We establish a similar result also for the solutions of the non-homogeneous version of the p-Laplace equation.     

基于均值-方差准则的保险公司最优投资策略

研究了保险公司在均值-方差准则下的最优投资问题,其中保险公司的盈余过程由带随机扰动的Cramer-Lundberg模型刻画,而且保险公司可将其盈余投资于无风险资产和一种风险资产.利用随机动态规划方法,通过求解相应的HJB方程,得到了均值方差模型的最优投资策略和有效前沿.最后,给出了数值算例说明扰动项对有效前沿的影响.     

Geometric existence theory for the control-affine nonlinear optimal regulator

For infinite horizon nonlinear optimal control problems in which the control term enters linearly in the dynamics and quadratically in the cost, well-known conditions on the linearised problem guarantee existence of a smooth globally optimal feedback solution on a certain region of state space containing the equilibrium point. The method of proof is to demonstrate existence of a stable Lagrangian manifold M and then construct the solution from M in the region where M has a well-defined projection onto state space. We show that the same conditions also guarantee existence of a nonsmooth viscosity solution and globally optimal set-valued feedback on a much larger region. The method of proof is to extend the construction of a solution from M into the region where M no-longer has a well-defined projection onto state space.     

Optimal proportional reinsurance and investment with minimum probability of ruin

The paper concerns a problem of optimal reinsurance and investment in order to minimizing the probability of ruin. In the whole paper, the cedent’s surplus is allowed to invest in a risk-free asset and a risky asset and the company’s risk is reduced through proportional reinsurance, while in addition the claim process is assumed to follow a Brownian motion with drift. By solving the corresponding Hamilton-Jacobi-Bellman equations, the optimal reinsurance-investment strategy is derived. The presented results generalize those by Taksar [1].     

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.     

Lower semicontinuous solutions of the Bellman equation for the minimum time problem

The minimum time problem associated with a nonlinear control system is considered, and the unicity of the lower semicontinuous solution of the corresponding Bellman equation is investigated. A main tool in our approach is the Kruzkov transformation that enables us to work on ⁿ –{0}, where {0} is the target set, instead of the unknown reachable set.This research was carried out while the first author was visiting the Department of Mathematics, University of Genova, Genova, Italy.     

On the local semiconcavity of the solutions of the eikonal equation

We show that if the Hamiltonian is locally semiconvex with respect to the state variables and strictly convex with respect to the gradient then every viscosity solution of the eikonal equation is locally semiconcave. Furthermore, in the 1D case, we show that every viscosity solution of the eikonal equation is semiconcave if and only if the Hamiltonian is Lipschitz continuous with respect to the state variable.     

Optimal control of stochastic functional differential equations with a bounded memory

This paper treats a finite time horizon optimal control problem in which the controlled state dynamics are governed by a general system of stochastic functional differential equations with a bounded memory. An infinite dimensional Hamilton–Jacobi–Bellman (HJB) equation is derived using a Bellman-type dynamic programming principle. It is shown that the value function is the unique viscosity solution of the HJB equation.