A unified framework for utility maximization problems: An Orlicz space approach

ABSTRACTIn 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.

In this paper, we study a class of integral equations with power nonlinearity on the entire real line. This class of equations arises in the -adic theory of open-closed strings. Using the method of successive approximations and justifying their convergence, we prove the existence of a nontrivial continuous odd bounded solution on the entire line. The asymptotic behavior of the solution is studied as the argument increases unboundedly.

This paper presents a three‐dimensional (3‐D: space‐time‐frequency) quasi proportional fairness (QPF)‐based radio resource allocation and interference coordination scheme in multicell long‐term evolution‐advanced downlink. We formulate the resource allocation and interference coordination problem as a QPF‐based utility maximisation problem, wherein, QPF reduces the complexity compared with the conventional proportional fairness. Interference has been minimised through time, frequency and space domain resource allocation. The proposed 3‐D scheme utilises two time slots for intercell interference management, spatial position‐based frequency resource allocation for intracell interference mitigation, and frequency domain orthogonality across user equipment (UE) and relays for inter‐UE and UE‐relays interference management. It has been shown that the relaxed non‐convex primal problem and the dual problem have zero duality gap under the time‐sharing and tolerable interference constraints. The relaxed optimal problem has been solved through the Lagrange dual method. Simulation results show the efficacy of the proposed 3‐D scheme compared with the recently presented three slots time domain interference coordination with round robin and proportional fairness time domain interference coordination schemes. It has been observed that the proposed heuristic 3D with equal power scheme exhibits same performance as the heuristic 3D with water‐filling power allocation and retains the desired optimal level for more than 60% of the average users' throughput. Copyright © 2016 John Wiley & Sons, Ltd.

This paper describes efficient algorithms for determining how buffer space should be allocated in a flow line. We analyze two problems: a primal problem, which minimizes total buffer space subject to a production rate constraint; and a dual problem, which maximizes production rate subject to a total buffer space constraint. The dual problem is solved by means of a gradient method, and the primal problem is solved using the dual solution. Numerical results are presented. Profit optimization problems are natural generalizations of the primal and dual problems, and we show how they can be solved using essentially the same algorithms.

In this paper, we consider the Lyapunov-type linear programming and its dual over symmetric cones. By introducing and characterizing the generalized inverse of Lyapunov operator in Euclidean Jordan algebras, we establish two kinds of Lyapunov-type Farkas’ lemmas to exhibit feasibilities of the corresponding primal and dual programming problems, respectively. As one of the main results, we show that the feasibilities of the primal and dual problems lead to the solvability of the primal problem and zero duality gap under some mild condition. In this case, we obtain that any solution to the pair of primal and dual problems is equivalent to the solution of the corresponding KKT system.

The paper studies connections between arbitrage and utility maximization in a discrete-time financial market. The market is incomplete. Thus one has several choices of equivalent martingale measures to price contingent claims. Davis determines a unique price for a contingent claim which is based on an optimal dynamic portfolio by use of a `marginal rate of substitution' argument. Here conditions will be given such that this price is determined by a martingale measure and thus by a consistent price system. The underlying utility function U is defined on the positive half-line. Then dynamic portfolios are admissible if the terminal wealth is positive. In case of the logarithmic utility function, the optimal dynamic portfolio is the numeraire portfolio.

3 - Further Topics in Linear Programming

We study regularity properties of the dynamic value functions of primal and dual problems of optimal investing for utility functions defined on the whole real line. Relations between decomposition terms of value processes of primal and dual problems and between optimal solutions of basic and conditional utility maximization problems are established. These properties are used to show that the value function satisfies a corresponding backward stochastic partial differential equation. In the case of complete markets we give conditions on the utility function when this equation admits a solution.

Portfolio Optimization Under Convex Incentive Schemes

We consider a firm purchasing a storable raw material commodity from a spot market with volatile commodity prices and the access to an associated financial derivatives market. The purchased commodity is processed into an end product with uncertain demand and lost sales. The firm aims to integrate the inventory replenishment and financial hedging decisions to maximize the mean‐variance of terminal wealth over a finite horizon. Recognizing time‐inconsistency of mean‐variance criteria, we employ the dynamic programming approach to obtain a time‐consistent policy. Assuming no arbitrage in financial market, we show that the mean‐variance utility functions under the time‐consistent policy have a recursive representation which enables us to readily characterize the structure of the time‐consistent policy. We analyze two types of hedging instruments, vanilla hedges and exotic hedges, and show that inventory and financial hedging decisions can be separated in the presence of forward contracts and a myopic state‐dependent base stock policy is optimal. The optimal hedging policy can be obtained by minimizing the variance of the hedging portfolio, the value of excess inventory and the profit‐to‐go as a function of future price. In the presence of a continuum of option strikes, we show how to construct custom exotic derivatives using forwards and options of all strikes to replicate the profit‐to‐go function. We then show the optimality of the time‐consistent policy under exotic hedge for the initial mean‐variance objective. We further investigate the dynamic interplay of inventories and financial hedge and show that they can be substitutes in a dynamic environment. Finally, we compare the performances in different hedging environments to discuss how financial hedges add value and provide a numerical study.

The purpose of this paper is to consider the optimal proportional reinsurance and investment strategies for an insurance company. The insurer’s surplus process is approximated by a Brownian motion with drift. The insurance company can purchase proportional reinsurance and invest the surplus in a financial market which includes one risk-free asset and one risky asset whose price is modeled by a CEV model. The primary problem is changed to the dual problem by implying Legendre transform. When the objective of the insurance company is to maximize the expected logarithmic utility from terminal wealth, the closed-form expressions for the optimal reinsurance-investment policy which is different to the Merton case to the primal optimal problem are obtained and numerical simulations are provided to demonstrate our results. Moreover, we find an interesting result that risk exposure is non-monotonic in the cost of reinsurance.

In this paper we study the problem of minimizing total weighted tardiness, a proxy for maximizing on-time delivery performance, on parallel nonidentical batch processing machines. We first formulate the (primal) problem as a nonlinear integer programming model. We then show that the primal problem can be solved exactly by solving a corresponding dual problem with a nonlinear relaxation. Since both the primal and the dual problems are NP-hard, we use genetic algorithms, based on random keys and multiple choice encodings, to heuristically solve them. We find that the genetic algorithms consistently outperform a standard mathematical programming package in terms of solution quality and computation time. We also compare the smaller problem instances to a breadth-first tree search algorithm that gives evidence of the quality of the solutions.

A neural net algorithm for multidimensional maximum entropy spectrum estimation

This chapter states the Monge–Kantorovich problem and provides the duality result in a fairly general setting. The primal problem is interpreted as the central planner's problem of determining the optimal assignment of workers to firms, while the dual problem is interpreted as the invisible hand's problem of obtaining a system of decentralized equilibrium prices. In general, the primal problem always has a solution (which means that an optimal assignment of workers to jobs exists), but the dual does not: the optimal assignment cannot always be decentralized by a system of prices. However, the cases where the dual problem does not have a solution are rather pathological, and in all of the cases considered in the rest of the book, both the primal and the dual problems have solutions.

In this paper, we consider the problem of robust optimization for a system with uncertainty of rank one. The main result is the canonical parameterization of all destabilizing uncertainties in the dual problem. The corresponding result in the rational case has been previously stated in terms of unstable zero-pole cancellations.In this paper the result is extended to the class of non-rational systems with continuous nominal factors. For non-rational systems the situation with the common zeros is more complicated. The nominal factors can contain a singular component and cannot be treated by unstable cancellations. We have shown that in the general case the common zeros of the plant factors are naturally replaced by a scalar function with the positive winding number. The result has certain similarities with the parameterization of the classical Nehari problem [9]. To illustrate the duality principle, the result is applied to a system with delay. The dual problem can be interpreted as the shortest distance from the nominal plant to all non-stabilizable plants in some metric that has a strong connection to and may be considered as a generalization of the υ-gap metric.