Portfolio optimization under shortfall risk constraint

This paper solves a utility maximization problem under utility-based shortfall risk constraint, by proposing an approach using Lagrange multiplier and convex duality. Under mild conditions on the asymptotic elasticity of the utility function and the loss function, we find an optimal wealth process for the constrained problem and characterize the bi-dual relation between the respective value functions of the constrained problem and its dual. This approach applies to both complete and incomplete markets. Moreover, the extension to more complicated cases is illustrated by solving the problem with a consumption process added. Finally, we give an example of utility and loss functions in the Black–Scholes market where the solutions have explicit forms.     

Conjugate duality in the optimization of the search for a target with generalized conditionally deterministic motion

A target moves in Euclideann-spaceR ⁿ according to the generalized conditionally deterministic law. The search density that accumulates on the target during its route determines the probability of detection. A necessary and sufficient condition for the search density (x, t) to be optimal is first represented, when there are two types of constraints for the search density: pointwise constraints and total-amount constraints. The second part consists of formulation of the dual problem with the aid of sensitivity parameters for the constraints. By using the dual functional, we obtain the maximal error from the minimum value of the primal objective functional for an arbitrary feasible . Finally, we study the discretized case, which is necessary for numerical calculations.     

Utility maximization problem with random endowment and transaction costs: when wealth may become negative

In this article, we study the problem of maximizing expected utility from the terminal wealth with proportional transaction costs and random endowment. In the context of the existence of consistent price systems, we consider the duality between the primal utility maximization problem and the dual one, which is set up on the domain of finitely additive measures. In particular, we prove duality results for utility functions supporting possibly negative values. Moreover, we construct a shadow market by the dual optimal process and consider the utility-based pricing for random endowment.     

Robust optimization with applications to game theory

In this article, we investigate robust optimization equilibria with two players, in which each player can neither evaluate his opponent's strategy nor his own cost matrix accurately while may estimate a bounded set of the strategy or cost matrix. We obtain a result that solving this equilibria can be formulated as solving a second-order cone complementarity problem under an ellipsoid uncertainty set or a mixed complementarity problem under a box uncertainty set. We present some numerical results to illustrate the behaviour of robust optimization equilibria.     

The path player game

We give an explicit PDE characterization for the solution of the problemof maximizing the utility of both terminal wealth and intertemporal consumption undermodel uncertainty. The underlying market model consists of a risky asset, whosevolatility and long-term trend are driven by an external stochastic factor process. Therobust utility functional is defined in terms of a HARA utility function with risk aversionparameter 0 < α < 1 and a dynamically consistent coherent risk measure, whichallows for model uncertainty in the distributions of both the asset price dynamics andthe factor process. Ourmethod combines recent results by Wittmüß (Robust optimizationof consumption with random endowment, 2006) on the duality theory of robustoptimization of consumption with a stochastic control approach to the dual problemof determining a ‘worst-case martingale measure’.     

Robust Optimal Portfolio Choice Under Markovian Regime-switching Model

We investigate an optimal portfolio selection problem in a continuous-time Markov-modulated financial market when an economic agent faces model uncertainty and seeks a robust optimal portfolio strategy. The key market parameters are assumed to be modulated by a continuous-time, finite-state Markov chain whose states are interpreted as different states of an economy. The goal of the agent is to maximize the minimal expected utility of terminal wealth over a family of probability measures in a finite time horizon. The problem is then formulated as a Markovian regime-switching version of a two-player, zero-sum stochastic differential game between the agent and the market. We solve the problem by the Hamilton-Jacobi-Bellman approach.      

Robust combinatorial optimization with variable cost uncertainty

We present in this paper a new model for robust combinatorial optimization with cost uncertainty that generalizes the classical budgeted uncertainty set. We suppose here that the budget of uncertainty is given by a function of the problem variables, yielding an uncertainty multifunction. The new model is less conservative than the classical model and approximates better Value-at-Risk objective functions, especially for vectors with few non-zero components. An example of budget function is constructed from the probabilistic bounds computed by Bertsimas and Sim. We provide an asymptotically tight bound for the cost reduction obtained with the new model. We turn then to the tractability of the resulting optimization problems. We show that when the budget function is affine, the resulting optimization problems can be solved by solving n+1$n+1$ deterministic problems. We propose combinatorial algorithms to handle problems with more general budget functions. We also adapt existing dynamic programming algorithms to solve faster the robust counterparts of optimization problems, which can be applied both to the traditional budgeted uncertainty model and to our new model. We evaluate numerically the reduction in the price of robustness obtained with the new model on the shortest path problem and on a survivable network design problem.     

不确定彩排时长下节目调度的鲁棒优化

实际节目彩排调度中,节目的表演时长受内外因素影响,具有不确定性。为了合理调度所有节目,控制演员的空闲时间,使得演员的总等待成本最小,采用了鲁棒优化方法进行研究。首先,建立了节目彩排调度的确定型模型;进一步,考虑节目表演时长的不确定性,采用有界区间描述节目表演时长并考虑决策者风险偏好,在确定型模型的基础上构建区间型两阶段鲁棒优化模型;接着,将鲁棒优化模型转化为0-1混合线性规划模型;最后,采用Matlab进行数值实验,结果表明决策者越偏好规避风险,演员的总等待成本越大。     

Applications of Toland's duality theory to nonconvex optimization problems

Through a suitable application of Toland's duality theory to certain nonconvex and nonsmooth problems one obtain an unbounded minimization problem with Fréchet:-differentiable cost function as dual problem and one can establish a gradient projection method for the solution of these problems.     

一类二元对策问题

讨论一类二元对策问题.在对策双方能力不同的情况下,推导对策双方获胜的充要条件和获胜方获胜的具体对策方案.文后的应用实例说明,本文所讨论的问题具有一定的应用价值.     

Market price-based convex risk measures: A distribution-free optimization approach

In Cont (2006) [1], a convex risk measure was proposed to measure the impact of uncertainty resulting from the misspecification of derivative models. Evaluation of the risk measures was illustrated on finite families of probability measures. In this paper, we consider the case of infinite families of measures that share common moments, e.g. mean and variance for European-style options. We show that the risk measure can still be efficiently evaluated based on semi-infinite programming. Examples are given that illustrate the benefits of evaluating the risk measure with infinite families of measures and shed light on the limitations of considering only finite families of measures.     

带交易费的终端资产和消费的期望贴现效用最大化

讨论带有一定比例交易费的关于贴现消费和贴现终端资产的期望效用最大化问题,在假定T时刻将资金全部转到银行中,利用对偶方法,求得最优消费c(t)和最大的终端财富X(T )．     

Adaptive control of a continuous-time portfolio and consumption model

In this paper, an adaptive control problem is formulated and solved using Merton's stochastic differential equation for the wealth in a portfolio selection and consumption model. Since the asset prices are assumed to satisfy a log normal distribution, it suffices to consider two assets. It is assumed that the drift parameter for the price of the risky asset is unknown. A recursive family of estimators for this unknown parameter is defined and is shown to converge almost surely to the true value of the parameter. The controls in the equation for the wealth are obtained from the optimal controls where the estimates of the unknown parameter are substituted for the unknown parameter.This research was partially supported by NSF Grant No. ECS-84-03286-A01.The authors wish to thank P. Varaiya for some useful comments on this paper.     

On the relationship between fenchel and Lagrange duality for optimization problems in general spaces

In this paper, the equivalence between a Fenchel and Lagrange duality theorem for optimization problems in dual pairs of real vector spaces is proved in a direct way.     

Optimal consumption and arbitrage in incomplete,finite state security markets

We study a consistent treatment for both the multi-period portfolio selection problem and the option attainability problem by a dual approach. We assume that time is discrete, the horizon is finite, the sample space is finite and the number of securities is less than that of the possible securities price transitions, i.e. an incomplete security market. The investor is prohibited from investing stocks more than given linear investment amount constraints at any time and he maximizes an expected additive utility function for the consumption process. First we give a set of budget feasibility conditions so that a consumption process is attainable by an admissible portfolio process. To establish this relation, we used an algorithmic approach which has a close connection with the linear programming duality. Then we prove the unique existence of a primal optimal solution from the budget feasibility conditions. Finally, we formulate a dual control problem and establish the duality between primal and dual control problems.We are grateful to the editor, Hiroshi Konno, and two anonymous referees for their valuable comments and constructive suggestions on this research. We are responsible for the remaining errors. The first author is supported in part by the fund endowed to the Research Association for Financial Engineering by Toyo Trust Bank Co. and Mito Shoken Co.     

Optimal consumption and investment with insurer default risk

We solve the optimal consumption and investment problem in an incomplete market, where borrowing constraints and insurer default risk are considered jointly. We derive in closed-form the optimal consumption and investment strategies. We find two main results by quantitative analysis. As insurer default risk increases, the proportion of wealth invested in stocks could increase when wealth is small, and decrease when wealth is large. As risk aversion increases, the voluntary annuity demand could increase when insurer default risk is low, and decrease when this risk is high.     

Optimal consumption and investment problem with random horizon in a BMAP model

In this paper, we consider the consumption and investment problem with random horizon in a Batch Markov Arrival Process (BMAP) model. The investor invests her wealth in a financial market consisting of a risk-free asset and a risky asset. The price processes of the riskless asset and the risky asset are modulated by a continuous-time Markov chain, which is the phase process of a BMAP. The possible consumption or investment are restricted to a sequence of random discrete time points which are determined by the same BMAP. The investor has only consumption opportunities at some of these random time points, has both consumption and investment opportunities at some other random time points, and can do nothing at the remaining random time points. The object of the investor is to select the consumption–investment strategy that maximizes the expected total discounted utility. The purpose of this paper is to analyze the impact of the consumption–investment opportunity and the economic state on the value functions and consumption–investment strategies. The general solution and the exact solution under the assumption that the consumption and the terminal wealth are evaluated by the power utility are obtained. Finally, a numerical example is presented.     

Dual extremum principles relating to an optimal filter

We obtain here a variational principle characterizing an optimal filter. This variational principle is dual to the one obtained by Berkovitz and Pollard in Ref. 2.The author wishes to thank the referee for his valuable suggestions.     

M-theory of matrix models

Small M-theories incorporate various models representing a unified family in the same way that the M-theory incorporates a variety of superstring models. We consider this idea applied to the family of eigenvalue matrix models: their M-theory unifies various branches of the Hermitian matrix model (including the Dijkgraaf-Vafa partition functions) with the Kontsevich τ-function. Moreover, the corresponding duality relations are reminiscent of instanton and meron decompositions, familiar from the Yang-Mills theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 179–192, February, 2007.     

具有共同冲击相依性的跳扩散金融市场中有着延迟和违约风险的鲁棒最优再保险和投资策略

在本文中,我们考虑跳扩散模型下具有延迟和违约风险的鲁棒最优再保险和投资问题,保险人可以投资无风险资产,可违约的债券和两个风险资产,其中两个风险资产遵循跳跃扩散模型且受到同种因素带来共同影响而相互关联.假设允许保险人购买比例再保险,特别地再保险保费利用均值方差保费原则来计算.在考虑与绩效相关的资本流入/流出下,保险公司的...