随机市场中均值-方差模型最优投资策略的时间不相容性及其修正

由于方差算子在动态规划意义下不可分,导致随机市场中多期均值一方差模型的最优投资策略不满足时间相容性,即Bellman最优性原理．为此,首先提出了随机市场中比Bellman最优性原理更弱的时间相容性,并证明在投资区间的任意中间时刻,当投资者的财富不超过某一给定的财富阈值时,最优投资策略满足弱时间相容性;当投资者的财富超过该阈值时,最优投资策略将不再是弱时间相容的,且导致投资者变为非理性,即他会同时极小化终期财富的均值和方差．在这种情形下,通过放松自融资约束,对最优投资策略进行了修正,使得其满足：修正策略可使投资者回归理性;相对于终期财富,修正策略可以获得与最优投资策略相同的均值和方差．在策略修正过程中,投资者可以从市场中获得一个严格正的现金流．这些结果表明修正策略要优于原最优投资策略,拓展了现有关于确定市场下多期均值．方差模型的求解以及策略时间相容性的结论．     

Stochastic optimization for the ruin probability

The Cramér‐Lundberg insurance model is studied where the risk process can be controlled by reinsurance and by investment in a financial market. The performance criterion is the ruin probability. The problem can be imbedded in the framework of discrete‐time stochastic dynamic programming. Basic tools are the Howard improvement and the verification theorem. Explicit conditions are obtained for the optimality of employing no reinsurance and of not investing in the market.     

Optimal production strategy under demand fluctuations: Technology versus capacity

This paper provides a comparative analysis of five possible production strategies for two kinds of flexibility investment, namely flexible technology and flexible capacity, under demand fluctuations. Each strategy is underpinned by a set of operations decisions on technology level, capacity amount, production quantity, and pricing. By evaluating each strategy, we show how market uncertainty, production cost structure, operations timing, and investment costing environment affect a firm’s strategic decisions. The results show that there is no sequential effect of the two flexibility investments. We also illustrate the different ways in which flexible technology and flexible capacity affect a firm’s profit under demand fluctuations. The results reveal that compared to no flexibility investment, flexible technology investment earns the same or a higher profit for a firm, whereas flexible capacity investment can be beneficial or harmful to a firm’s profit. Moreover, we prove that higher flexibility does not guarantee more profit. Depending on the situation, the optimal strategy can be any one of the five possible strategies. We also provide the optimality conditions for each strategy.     

Discrete approximation of relaxed optimal control problems

We consider a general nonlinear optimal control problem for systems governed by ordinary differential equations with terminal state constraints. No convexity assumptions are made. The problem, in its so-called relaxed form, is discretized and necessary conditions for discrete relaxed optimality are derived. We then prove that discrete optimality [resp., extremality] in the limit carries over to continuous optimality [resp., extremality]. Finally, we prove that limits of sequences of Gamkrelidze discrete relaxed controls can be approximated by classical controls.     

Necessary global optimality conditions for nonlinear programming problems with polynomial constraints

In this paper, we develop necessary conditions for global optimality that apply to non-linear programming problems with polynomial constraints which cover a broad range of optimization problems that arise in applications of continuous as well as discrete optimization. In particular, we show that our optimality conditions readily apply to problems where the objective function is the difference of polynomial and convex functions over polynomial constraints, and to classes of fractional programming problems. Our necessary conditions become also sufficient for global optimality for polynomial programming problems. Our approach makes use of polynomial over-estimators and, a polynomial version of a theorem of the alternative which is a variant of the Positivstellensatz in semi-algebraic geometry. We discuss numerical examples to illustrate the significance of our optimality conditions.     

Minimization of risks in defined benefit pension plan with time‐inconsistent preferences

In this paper, we investigate the defined benefit pension plan, where the object of the manager is to minimise the contribution rate risk and the solvency risk by considering a quadratic performance criterion. To incorporate some well‐documented behavioural features of human beings, we consider the situation where the discounting is non‐exponential. It leads to a time‐inconsistent control problem in the sense that the Bellman optimality principle does no longer hold. In our model, we assume that the benefit outgo is constant, and the pension fund can be invested in a risk‐free asset and a risky asset whose return follows a geometric Brownian motion. We characterise the time‐consistent strategies and value function in terms of the solution of a system of integral equations. The existence and uniqueness of the solution is verified, and the approximation of the solution is obtained. Some numerical results of the equilibrium contribution rate and equilibrium investment policy are presented for three types of discount functions. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal prevention strategies in the classical risk model

In this paper, we propose and study a first risk model in which the insurer may invest into a prevention plan which decreases claim intensity. We determine the optimal prevention investment for different risk indicators. In particular, we show that the prevention amount minimizing the ruin probability maximizes the adjustment coefficient in the classical ruin model with prevention, as well as the expected dividends until ruin in the model with dividends. We also show that the optimal prevention strategy is different if one aims at maximizing the average surplus at a fixed time horizon. A sensitivity analysis is carried out. We also prove that our results can be extended to the case where prevention starts to work only after a minimum prevention level threshold.     

Decision model and analysis for investment interest expense deduction and allocation

Investment income tax planning requires informed, strategic choices. One must determine the amount of qualified dividends and net long-term capital gain to be included in investment income (against which investment interest expense can be deducted). This choice also determines the residual qualified dividends and net long-term capital gain which enjoy a reduced tax rate. Another important decision is whether all or some of this interest expense should be deducted in the current year or carried forward. This paper puts forward a new approach to formulate these questions as a generalized resource allocation problem which permits analysis of the interdependence between, and the tax consequences of, the above decisions. The commonly used approach – deducting investment interest expense sooner rather than later – we consider myopic since the benefit of deferring some of the deduction is not leveraged. Presented here is a tax planning guideline (a necessary and sufficient condition for optimality) to realize a more forward-looking strategy. We also show that, for certain income structures, the tax savings by deducting a one-dollar investment interest expense may be more than the tax rate on the dollar of investment income that is offset.     

Managing contribution and capital market risk in a funded public defined benefit plan: Impact of CVaR cost constraints

Using a Monte Carlo framework, we analyze the risks and rewards of moving from an unfunded defined benefit pension system to a funded plan for German civil servants, allowing for alternative strategic contribution and investment patterns. In the process we integrate a Conditional Value at Risk (CVaR) restriction on overall plan costs into the pension manager’s objective of controlling contribution rate volatility. After estimating the contribution rate that would fully fund future benefit promises for current and prospective employees, we identify the optimal contribution and investment strategy that minimizes contribution rate volatility while restricting worst-case plan costs. Finally, we analyze the time path of expected and worst-case contribution rates to assess the chances of reduced contribution rates for current and future generations. Our results show that moving toward a funded public pension system can be beneficial for both civil servants and taxpayers.     

Analysis for some properties of discrete time Markov decision processes

This paper investigates properties of the optimality equation and optimal policies in discrete time Markov decision processes with expected discounted total rewards under weak conditions that the model is well defined and the optimality equation is true. The optimal value function is characterized as a solution of the optimality equation and the structure of optimal policies is also given.     

Weak and Strong Optimality Conditions for Constrained Control Problems with Discontinuous Control

In recent years, sufficient optimality criteria and solution stability in optimal control have been investigated widely and used in the analysis of discrete numerical methods. These results were concerned mainly with weak local optima, whereas strong optimality has been considered often as a purely theoretical aspect. In this paper, we show via an example problem how weak the weak local optimality can be and derive new strong optimality conditions. The criteria are suitable for practical verification and can be applied to the case of discontinuous controls with changes in the set of active constraints.     

带有通胀风险的退休后最优投资-年金化时刻决策

研究了带通货膨胀的确定缴费养老计划退休后最优投资-年金化决策。假设通货膨胀过程是一个随机过程,建立了真实财富的波动过程。先相对固定年金化时刻,采取目标定位型模型,预设未来各时期的投资目标,利用贝尔曼优化原理,得到从退休时刻到相对固定年金化时刻之间的最优投资策略。接着建立了最优年金化时刻的评估标准,最优的年金化时刻使得年金化前后的累加消费折现均值得到最大。证明了在随机通货膨胀的假设下,传统的自然投资目标不存在;当随机通胀过程退化到确定过程时,求出了自然投资目标的显式表达式,并且在这两种情况下,分析了通胀情况对最优投资策略的影响。最后利用数值分析手段, 研究了通货膨胀、风险偏好、折现率对最优年金化时刻的影响。     

On the No-Gap Second-Order Optimality Conditions for a Discrete Optimal Control Problem with Mixed Constraints

Motivated by our recent works on optimality conditions in discrete optimal control problems under a nonconvex cost function, in this paper, we study second-order necessary and sufficient optimality conditions for a discrete optimal control problem with a nonconvex cost function and state-control constraints. By establishing an abstract result on second-order optimality conditions for a mathematical programming problem, we derive second-order necessary and sufficient optimality conditions for a discrete optimal control problem. Using a common critical cone for both the second-order necessary and sufficient optimality conditions, we obtain “no-gap” between second-order optimality conditions.     

Nonconvex and relaxed infinite-horizon optimal control problems

In this paper, we consider the Lagrange problem of optimal control defined on an unbounded time interval in which the traditional convexity hypotheses are not met. Models of this form have been introduced into the economics literature to investigate the exploitation of a renewable resource and to treat various aspects of continuous-time investment. An additional distinguishing feature in the models considered is that we do not assume a priori that the objective functional (described by an improper integral) is finite, and so we are led to consider the weaker notions of overtaking and weakly overtaking optimality. To treat these models, we introduce a relaxed optimal control problem through the introduction of chattering controls. This leads us naturally to consider the relationship between the original problem and the convexified relaxed problem. In particular, we show that the relaxed problem may be viewed as a limiting case for the original problem. We also present several examples demonstrating the applicability of our results.     

A modified active set algorithm for transportation discrete network design bi-level problem

Transportation discrete network design problem (DNDP) is about how to modify an existing network of roads and highways in order to improve its total system travel time, and the candidate road building or expansion plan can only be added as a whole. DNDP can be formulated into a bi-level problem with binary variables. An active set algorithm has been proposed to solve the bi-level discrete network design problem, while it made an assumption that the capacity increase and construction cost of each road are based on the number of lanes. This paper considers a more general case when the capacity increase and construction cost are specified for each candidate plan. This paper also uses numerical methods instead of solvers to solve each step, so it provides a more direct understanding and control of the algorithm and running procedure. By analyzing the differences and getting corresponding solving methods, a modified active set algorithm is proposed in the paper. In the implementation of the algorithm and the validation, we use binary numeral system and ternary numeral system to avoid too many layers of loop and save storage space. Numerical experiments show the correctness and efficiency of the proposed modified active set algorithm.     

On the bailout dividend problem with periodic dividend payments for spectrally negative Markov additive processes

This paper studies the bailout optimal dividend problem with regime switching under the constraint that dividend payments can be made only at the arrival times of an independent Poisson process while capital can be injected continuously in time. We show the optimality of the regime-modulated Parisian-classical reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. In order to verify the optimality, first we study an auxiliary problem driven by a single spectrally negative Lévy process with a final payoff at an exponential terminal time and characterize the optimal dividend strategy. Then, we use the dynamic programming principle to transform the global regime-switching problem into an equivalent local optimization problem with a final payoff up to the first regime switching time. The optimality of the regime modulated Parisian-classical barrier strategy can be proven by using the results from the auxiliary problem and approximations via recursive iterations.     

Inventory control with fixed cost and price optimization in continuous time

We continue to study the problem of inventory control, with simultaneous pricing optimization in continuous time. In our previous paper [8], we considered the case without set up cost, and established the optimality of the base stock-list price (BSLP) policy. In this paper we consider the situation of fixed price. We prove that the discrete time optimal strategy (see [11]), i.e., the (s, S, p) policy can be extended to the continuous time case using the framework of quasi-variational inequalities (QVIs) involving the value function. In the process we show that an associated second order, nonlinear two-point boundary value problem for the value function has a unique solution yielding the triplet (s, S, p). For application purposes the explicit knowledge of this solution is needed to specify the optimal inventory and pricing strategy. Se- lecting a particular demand function we are able to formulate and implement a numerical algorithm to obtain good approximations for the optimal strategy.     

Global Quadratic Minimization over Bivalent Constraints: Necessary and Sufficient Global Optimality Condition

In this paper, we establish global optimality conditions for quadratic optimization problems with quadratic equality and bivalent constraints. We first present a necessary and sufficient condition for a global minimizer of quadratic optimization problems with quadratic equality and bivalent constraints. Then we examine situations where this optimality condition is equivalent to checking the positive semidefiniteness of a related matrix, and so, can be verified in polynomial time by using elementary eigenvalues decomposition techniques. As a consequence, we also present simple sufficient global optimality conditions, which can be verified by solving a linear matrix inequality problem, extending several known sufficient optimality conditions in the existing literature.     

A Discrete Laplace–Beltrami Operator for Simplicial Surfaces

We define a discrete Laplace–Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace–Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa’s Theorem we show that, as claimed, Musin’s harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces. Research for this article was supported by the DFG Research Unit 565 “Polyhedral Surfaces” and the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.