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.     

An analytical framework for aggregating multiattribute utility functions

This paper proposes a procedure for aggregating individual cardinal utility functions into a social utility function that represents the preferences of all the individuals as a whole. The procedure is non-interactive and is based upon the determination of the utility consensus values. This is accomplished by minimizing a distance function model that is transformed into an Archimedean goal programming problem. The procedure is applied to a general group multilinear utility function.     

Imbedding methods for the integral equations for the optimal control of time-lag systems,I: Parameter dependence

The problem of the optimal control of systems with time lags can be reduced to a Fredholm integral equation. In this paper, it is shown that the Fredholm resolvent satisfies an initial-value problem in which a parameter serves as the independent variable. This reduction may be of computational utility.This work was partially supported by Sigma Delta Epsilon—Graduate Women in Science and NIH Grant No. GM-23732.     

A quasi-variational approach to a competitive economic equilibrium problem without strong monotonicity assumption

This paper is focused on the investigation of the Walrasian economic equilibrium problem involving utility functions. The equilibrium problem is here reformulated by means of a quasi-variational inequality problem. Our goal is to give an existence result without assuming strong monotonicity conditions. To this end, we make use of a perturbation procedure. In particular, we will consider suitable perturbed utility functions whose gradient satisfies a strong monotonicity condition and whose associated equilibrium problem admits a solution. Then, we will prove that the limit solution solves the unperturbed problem. We stress out that our result allows us to consider a wide class of utility functions in which the Walrasian equilibrium problem may be solved.     

An Algorithm for Portfolio Optimization with Variable Transaction Costs,Part 1: Theory

A portfolio optimization problem consists of maximizing an expected utility function of n assets. At the end of a typical time period, the portfolio will be modified by buying and selling assets in response to changing conditions. Associated with this buying and selling are variable transaction costs that depend on the size of the transaction. A straightforward way of incorporating these costs can be interpreted as the reduction of portfolios’ expected returns by transaction costs if the utility function is the mean-variance or the power utility function. This results in a substantially higher-dimensional problem than the original n-dimensional one, namely (2K+1)n-dimensional optimization problem with (4K+1)n additional constraints, where 2K is the number of different transaction costs functions. The higher-dimensional problem is computationally expensive to solve. This two-part paper presents a method for solving the (2K+1)n-dimensional problem by solving a sequence of n-dimensional optimization problems, which account for the transaction costs implicitly rather than explicitly. The key idea of the new method in Part 1 is to formulate the optimality conditions for the higher-dimensional problem and enforce them by solving a sequence of lower-dimensional problems under the nondegeneracy assumption. In Part 2, we propose a degeneracy resolving rule, address the efficiency of the new method and present the computational results comparing our method with the interior-point optimizer of Mosek. This research was supported by the National Science and Engineering Research Council of Canada and the Austrian National Bank. The authors acknowledge the valuable assistance of Rob Grauer and Associate Editor Franco Giannessi for thoughtful comments and suggestions.     

Modelling the Process of Multiattribute Choice

In most of the multiattribute utility models it is assumed that the analytical forms of partial and overall utility functions are known, and hence the problem becomes one of estimating the values of model parameters. This assumption is rather restrictive since one may not possess prior knowledge as to the analytical forms of such functions. The present paper is an attempt to eliminate some of the shortcomings of the existing models by proposing two new formulations of the multiattribute choice problem without a priori limitation to any strict form of partial utility function.     

Consumption-Investment Problem with Subsistence Consumption,Bankruptcy, and Random Market Coefficients

We consider a general continuous-time finite-horizon single-agent consumption and portfolio decision problem with subsistence consumption and value of bankruptcy. Our analysis allows for random market coefficients and general continuously differentiable concave utility functions. We study the time of bankruptcy as a problem of optimal stopping, and succeed in obtaining explicit formulas for the optimal consumption and wealth processes in terms of the optimal bankruptcy time. This paper extends the results of Karatzas, Lehoczky, and Shreve (Ref. 1) on the maximization of expected utility from consumption in a financial market with random coefficients by incorporating subsistence consumption and bankruptcy. It also addresses the random coefficients and finite-horizon version of the problem treated by Sethi, Taksar, and Presman (Ref. 2). The mathematical tools used in our analysis are optimal stopping, stochastic control, martingale theory, and Girsanov change of measure.     

Designing a mixed-model,open-station assembly line using mixed-integer programming

An open-station assembly line that manufactures mixed models of sheet metal cabinets is considered in this research. The problem minimizes the total cost of the idle and utility times incurred in an assembly line with different line parameters (such as launch interval, station length, starting point of work, upstream walk, locus of the operator's movement, etc.) and operation sequences of the mixed models. An open-station system plays a significant controlling role in determining the optimal line parameters that minimize the total cost of idle and utility times in a mixed-model assembly line. Thus, a mixed-integer programming model for an open-station system is developed here to determine line parameters optimally. The model is tested on a three-station mixed-model line, which is a partial representation of a complete long assembly line. This research obtained a set of line parameters that minimize the total cost of idle and utility times optimally. Results indicate that the minimum total cost of idle and utility times in an open-station system decreases with line length. Other results pertinent to the line design are also demonstrated.     

多周期公用工程系统运行的模型,优化方法与应用

针对多周期公用工程系统的运行优化问题，考虑了设备的启停费用的情况下。建立了混合整数非线性规划模型并证明了最优解的存在性。针对该运行优化问题本将其分解成若干子问题，然后利用改进的Hooke-Jeeves优化算法求解每个子问题。应用于具体实例，其数值结果与其它方法得到的相比。运行时间短，且更适合多周期公用工程问题的求解。     

On minkowski metric and weighted tchebyshev norm in vector optimization ∗

The typical approach in solving vector optimization problems is to scalarize the vector cost function into a single cost function by means of some utility or value function. A very large class of utility function is given by the Minkowski’s metric proposed by Charnes and Cooper in the context of goal programming. This includes the special case of linear scalarization and the weighted Tchebyshev norm. We shall furnish a rigorous justification that there is no equivalent relationship between the general vector optimization problem and scalarized optimization problems using any Minkowski’s metric utility function. Furthermore, we also show that the weighted Tchebyshev norm is, in some sense, the best amongst the class of Minkowski’s metric utility functions since it is the only scalarization method which yields an equivalence relation between the weak vector optimization problem and a set of scalar optimization problems, without any convexity assumption     

Event risk, contingent claims and the temporal resolution of uncertainty

We study the pricing and hedging of contingent claims that are subject to Event Risk which we define as rare and unpredictable events whose occurrence may be correlated to, but cannot be hedged perfectly with standard marketed instruments. The super-replication costs of such event sensitive contingent claims (ESCC), in general, provide little guidance for the pricing of these claims. Instead, we study utility based prices under two scenarios of resolution of uncertainty for event risk: when the event is continuously monitored, or when it is revealed only at the payment date. In both cases, we transform the incomplete market optimal portfolio choice problem of an agent endowed with an ESCC into a complete market problem with a state and possibly path-dependent utility function. For negative exponential utility, we obtain an explicit representation of the utility based prices under both information resolution scenarios and this in turn leads us to a simple characterization of the early resolution premium. For constant relative risk aversion utility functions we propose a simple numerical scheme and study the impact of size of the position, wealth and expected return on these prices.     

Utility maximization with partial information: Hamilton-Jacobi-Bellman equation approach

This paper deals with the problem of maximizing the expected utility of the terminal wealth when the stock price satisfies a stochastic differential equation with instantaneous rates of return modelled as an Ornstein-Uhlenbeck process. Here, only the stock price and interest rate can be observable for an investor. It is reduced to a partially observed stochastic control problem. Combining the filtering theory with the dynamic programming approach, explicit representations of the optimal value functions and corresponding optimal strategies are derived. Moreover, closed-form solutions are provided in two cases of exponential utility and logarithmic utility. In particular, logarithmic utility is considered under the restriction of short-selling and borrowing.      

Insider Models with Finite Utility in Markets with Jumps

In this article we consider, under a Lévy process model for the stock price, the utility optimization problem for an insider agent whose additional information is the final price of the stock blurred with an additional independent noise which vanishes as the final time approaches. Our main interest is establishing conditions under which the utility of the insider is finite. Mathematically, the problem entails the study of a “progressive” enlargement of filtration with respect to random measures. We study the jump structure of the process which leads to the conclusion that in most cases the utility of the insider is finite and his optimal portfolio is bounded. This can be explained financially by the high risks involved in models with jumps.     

Utility maximization under risk constraints and incomplete information for a market with a change point

In this article, we consider an optimization problem of expected utility maximization of continuous-time trading in a financial market. This trading is constrained by a benchmark for a utility-based shortfall risk measure. The market consists of one asset whose price process is modelled by a Geometric Brownian motion where the market parameters change at a random time. The information flow is modelled by initially and progressively enlarged filtrations which represent the knowledge about the price process, the Brownian motion and the random time. We solve the maximization problem and give the optimal terminal wealth depending on these different filtrations for general utility functions by using martingale representation results for the corresponding filtration.     

Optimal investment strategies with a reallocation constraint

We study a Merton type optimization problem under a reallocation constraint. Under this restriction, the stock holdings can not be liquidated faster than a certain rate. This is a common restriction in certain type of investment firms. Our main objective is to study the large time optimal growth rate of the expected value of the utility from wealth. We also consider a discounted infinite horizon problem as a step towards understanding the first problem. A numerical study is done by solving the dynamic programming equations. Under the assumption of a power utility function, an appropriate dimension reduction argument is used to reduce the original problem to a two dimensional one in a bounded domain with convenient boundary conditions. Computation of the optimal growth rate introduces additional numerical difficulties as the straightforward approach is unstable. In this direction, new analytical results characterizing the growth rate as the limit of a sequence of finite horizon problems with continuously derived utility are proved.     

状态相依效用下的超额损失再保险——投资策略

假设保险公司的盈余过程和金融市场的资产价格过程均由可观测的连续时间马尔科夫链所调节, 以最大化终端财富的状态相依的期望指数效用为目标, 研究了保险公司的超额损失再保险-投资问题. 运用动态规划方法, 得到最优再保险-投资策略的解析解以及最优值函数的半解析式. 最后, 通过数值例子, 分析了模型各参数对最优值函数和最优策略的影响.     

Optimal Investment Strategies for an Insurer with SAHARA Utility

In this paper, we consider the optimal investment strategy which maximizes the utility of the terminal wealth of an insurer with SAHARA utility functions. This class of utility functions has non-monotone absolute risk aversion, which is more flexible than the CARA and CRRA utility functions. In the case that the risk process is modeled as a Brownian motion and the stock process is modeled as a geometric Brownian motion, we get the closed-form solutions for our problem by the martingale method for both the constant threshold and when the threshold evolves dynamically according to a specific process. Finally, we show that the optimal strategy is state-dependent.     

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??In this paper, we consider the optimal investment strategy which maximizes the utility of the terminal wealth of an insurer with SAHARA utility functions. This class of utility functions has non-monotone absolute risk aversion, which is more flexible than the CARA and CRRA utility functions. In the case that the risk process is modeled as a Brownian motion and the stock process is modeled as a geometric Brownian motion, we get the closed-form solutions for our problem by the martingale method for both the constant threshold and when the threshold evolves dynamically according to a specific process. Finally, we show that the optimal strategy is state-dependent.     

The multi-armed bandit, with constraints

Presented in this paper is a self-contained analysis of a Markov decision problem that is known as the multi-armed bandit. The analysis covers the cases of linear and exponential utility functions. The optimal policy is shown to have a simple and easily-implemented form. Procedures for computing such a policy are presented, as are procedures for computing the expected utility that it earns, given any starting state. For the case of linear utility, constraints that link the bandits are introduced, and the constrained optimization problem is solved via column generation. The methodology is novel in several respects, which include the use of elementary row operations to simplify arguments.     

A generalized design approach to solution of the non-convex quadratic programming problem

This paper outlines a generalized, systematic design approach to solution of the non-convex quadratic programming problem. It is based on a previous formulation of utility of a general system in terms of efficiency and robustness of the system. The approach is to introduce a robustness term of arbitrary magnitude into the design utility function. Mathematically, this makes the problem convex. From a design approach, it yields a more general solution allowing specialization to proceed by decreasing robustness (on an initially convex utility surface in the feasible design space) until the region of the overall optimum is approached. The approach is mathematically related to the Metropolis technique of simulated annealing but a more systematic (less random) solution process is used. It is analogous also to the heuristic technique of Burkard and Bonniger. These two previous techniques are the most effective so far reported for the quadratic programming problem. The robustness approach provides an underpinning for each and opens up further solution options. Applications include layout of buildings and other constructed facilities and information technology layout problems.