Recently, Iwamoto, Kimura, and Ueno proposed dynamic dualization to present dual problems for unconstrained optimization problems whose objective function is a sum of squares. The aim of this paper is to show that dynamic dualization works well for unconstrained problems whose objective function is a sum of convex functions. Further we give another way to get dual problems, which is based on the infimal convolution. In both approaches we make clear the assumption for duality to hold.
This paper deals with a refinement of the Laplace-Carson transform (LCT) approach to option pricing, with a special emphasis on valuing defaultable and non-callable convertible bonds (CBs), but not limited to it. What we are actually aiming at is refining the plain LCT approach to meet possibly general American derivatives. The setup is a standard Black-Scholes-Merton framework where the underlying firm value evolves according to a geometric Brownian motion. The valuation of CBs can be formulated as an optimal stopping problem, due to the possibility of voluntary conversion prior to maturity. We begin with the plain LCT approach that generates a complex solution with little prospect of further analysis. To improve this solution, we introduce the notion of premium decomposition, which separates the CB value into the associated European CB value and an early conversion premium. By the LCT approach combined with the premium decomposition, we obtain a much simpler and closed-form solution for the CB value and an optimal conversion boundary. By virtue of the simplified solution, we can easily characterize asymptotic properties of the early conversion boundary. Finally, we show that our refined LCT approach is broadly applicable to a more general class of claims with optimal stopping structure.
We consider fuzzy sets on a metric, vector, or normed space. It is not assumed that the fuzzy sets have compact supports. In the present paper, a fuzzy distance and a fuzzy norm are proposed in order to measure the difference between two fuzzy sets, and their fundamental properties are investigated. Their definitions are based on Zadeh's extension principle. Although they are different from the classical ones based on the Hausdorff metric, they are suitable for data containing uncertainty or vagueness. The obtained results can be expected to be useful for analyzing such data when the data are represented as fuzzy sets.
It is shown that the Fibonacci sequence is optimal for two quadratic programming problems (maximization and minimization) under semi-Fibonacci constraints. The two conditional (primal) problems have their unconditional (dual) problems. The optimal solution is characterized by the Fibonacci number. Both pairs of primal and dual problems are mutually derived through three methods — dynamic, plus-minus and inequality —.
Recently, a seating position can be often selected when a plane or bullet train ticket is reserved. Specially, for theater and stadium, it is important to decide how to assign reservations to seats. This paper proposes a dynamic model where seats' resources are located at a single line with considering seats position that have already been assigned. An analysis has been conducted and the results show that, 1) optimal policy for an arriving request is to allocate it to one side of the edges of the adjacent vacancies, 2) if all of the resources are vacant at beginning time for booking, then the model corresponds to a single-leg model with multiple seat bookings and single fare class in Lee and Hersh (1993), 3) it is not necessarily optimal that a request is allocated to the less adjacent seats' vacancy. Finally, this paper proposes an algorithm to solve the optimal policy using above results and conducts numerical examples.
The particle survival model, which was originally proposed to analyze the dynamics of species' coexistence, has surprisingly found to be related to a non-homogeneous Poisson process. It is also well known that successive record values of independent and identically distributed sequences have the spatial distribution of such processes. In this paper, we show that the particle survival model and the record value process are indeed equivalent. Further, we study their application to determine the optimal strategy for placing selling orders on stock exchange limit order books. Our approach considers the limit orders as particles, and assumes that the other traders have zero intelligence.
The purpose of this study is to consider the problem of finding a guaranteed way of winning a certain two-player combinatorial game of perfect knowledge from the standpoint of mutually dependent decision processes (MDDPs). Our MDDP model comprises two one-stage deterministic decision processes. Each decision process expresses every turn of a player. We analyze a MDDP problem in which the length of turns taken by a player is minimized, allowing him to win regardless of the decisions made by his opponent. The model provides a formulation for finding the shortest guaranteed strategy. Although computational complexity remains, the concept introduced in this paper can also be applied to other two-player combinatorial games of perfect knowledge.
Any positive semi-definite function defined on Z (resp. R) can be represented as the Fourier transform of a positive Radon measure on T (resp. R). We give a proof of this celebrated result due to Herglotz and Bochner from the viewpoint of Schwartz's theory of distributions.
This paper studies the relation between a given nondeterministic discrete decision process (nd-ddp) and a nondeterministic sequential decision process (nd-sdp), which is a finite nondeterministic automaton with a cost function, and its subclasses (nd-msdp, nd-pmsdp, nd-smsdp). We show super-strong representation theorems for nd-sdp and its subclasses, for which the functional equations of nondeterministic dynamic programming are obtainable. The super-strong representation theorems provide necessary and sufficient conditions for the existence of the nd-sdp and its subclasses with the same set of feasible policies and the same cost value for every feasible policy as the given process nd-ddp.
This paper deals with security games which would be found around our lives. In a facility represented by a network, several types of invaders/attackers conflict with security guards/defenders who have also several security teams. The attacker chooses an invasion path to move along. He incurs some attrition by the conflict on arcs but surviving attackers give damage to the facility on his invasion route while the defender tries to minimize the damage by intercepting the attacker by a limited number of guards. The defender takes a randomized plan with respect to the adoption of each security team and the deployment of guards. Since the attacker know the defender's randomized plan before his decision making, the security problem is modeled by a Stackelberg game with the superiority of the attacker on information acquisition to the defender. There has been no research on the security game with multiple types of players modeled on a network, which explicitly takes account attrition on players. By some numerical examples, we investigate the best configuration of staff numbers in security teams and some characteristics of optimal defense to mitigate the damage caused by the attackers.
In this paper, we consider the pricing decision of a retailer who experiences peak demand for a product during a given time interval and wishes to stabilize the demand by adjusting the sales price. The stabilization of demand brings about desirable outcomes such as a reduction in the need for capacity investment and improves the production efficiency in the supply chain. We establish a continuous-time model to analyze the effect of dynamic pricing on peak demand. We find that a closed-form optimal pricing policy minimizes the difference between the actual demand and target level. It is shown that the dynamic pricing not only reduces peak demand but also mitigates fluctuations in the peak demand. Using electricity consumption data as a case study, we show that the proposed pricing policy is effective for reducing the mean peak demand compared to a constant pricing policy.
Project risk is an uncertain event that causes positive or negative effects on the project objectives in relation to the cost, time, quality and so on to complete the project. Project risk management is the set of processes of identifying, analyzing and responding to project risks. For example, the project risk management includes the process of eliminating the project risks from the project to complete any activities in the project by the specified day. In terms of not only the risk but also the time, many researches have been done. Especially, as for the time, there are many researches on CPM and PERT, which use mathematical techniques. However, few researchers discuss the effectiveness of project risk responses to deal with project risks for making a success of the project. In this paper, we propose a new mathematical model of the project risk responses. And, with our proposing model, we show how to calculate the effectiveness of project risk responses quantitatively. Moreover, we can decide quantitatively which project risk response should be executed by the consequences of the above calculation.
We provide a sufficient condition for the existence of a Markov perfect equilibrium for pure strategies in a class of Markov games where each stage has strategic complementarities. We assume that both the sets of actions for all players and the set of states are finite and that the horizon is also finite, while the past studies examined Markov games with infinite horizons where the sets of actions and states are assumed to be infinite. We give an elementary proof of the existence and apply the result to a game of Bertrand oligopoly with investment.
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