A robust model for a leader–follower competitive facility location problem in a discrete space

This paper aims at determining the optimal locations for the leader’s new facilities under the condition that the number of the follower’s new facilities is unknown for the leader. The leader and the follower have some facilities in advance. The first competitor, the leader, opens p new facilities in order to increase her own market share. On the other hand, she knows that her competitor, the follower, will react to her action and locate his new facilities as well. The number of the follower’s new facilities is unknown for the leader but it is assumed that the leader knows the probability of opening different numbers of the follower’s new facilities. The leader aims at maximizing her own market share after the follower’s new facilities entry. The follower’s objective is also to maximize his own market share. Since the number of the follower’s new facilities is unknown for leader, “Robust Optimization” is used for maximizing the leader’s market share and making the obtained results “robust” in various scenarios in terms of different numbers of the follower’s new facilities. The optimal locations for new facilities of both the leader and the follower are chosen among pre-determined potential locations. It is assumed that the demand is inelastic. The customers probabilistically meet their demands from all different facilities and the demand level which is met by each facility is computed by Huff rule. The computational experiments have been applied to evaluate the efficiency of the proposed model.     

Nash bargaining and leader–follower models in water allocation: Application to the Zarrinehrud River basin,Iran

Mathematical models for conflict resolution are very important in integrated water resources and environmental management. This study proposes a new methodology to resolve conflicts among different water users and water suppliers while considering environmental requirements and the system’s constraints. A two-level leader–follower model is applied to maximize the net benefit with the Iran Water Resources Management Company as the leader and agricultural, domestic, and industrial users as followers subject to the system’s constraints. As a comparison, the Nash bargaining solution is also used to find a solution when simultaneous moves are assumed by the participants. The suggested method is then applied to the real case of the Zarrinehrud River basin that is one of the areas facing water shortages in Iran. For the actual optimization, Genetic Algorithm is used in order to avoid local optimum. As the contribution of this study, the results show that benefits for the leader in the leader–follower model increased in comparison with the Nash bargaining solutions.     

Gibbs–Markov–Young structures with (stretched) exponential tail for partially hyperbolic attractors

In this work we extend the results obtained by Gouëzel in [12] to partially hyperbolic attractors. We study a forward invariant set K on a Riemannian manifold M   whose tangent space splits as dominated decomposition _TKM=E^cu⊕E^s$T_{K}M=E^{cu}\oplus E^s$, for which the center-unstable direction E^cu$E^{cu}$ is non-uniformly expanding on some local unstable disk. We prove that the (stretched) exponential decay of recurrence times for an induced scheme can be deduced under the assumption of (stretched) exponential decay of the time that typical points need to achieve some uniform expanding in the center-unstable direction. As an application of our results we obtain exponential decay of correlations and exponential large deviations for a class of partially hyperbolic diffeomorphisms considered in [1].     

The finite–horizon version for a partially–observed stochastic control problem of benesš & rishel

A Bayesian adaptive control problem with several interesting features, due to Bene? and Rishel, was treated as a stochastic control problem with partial observations – and on an infinite horizon with discounting – in thepapers [2] and [10]. We discuss here in full detail the finite–horizon version of that problem, by solving fairly explicitly the associated, fully nonlinear and degenerate, Hamilton-Jacobi-Bellman equation of parabolic type     

A game theory approach in seller–buyer supply chain

In this paper, several seller–buyer supply chain models are proposed which incorporate both cost factors as well as elements of competition and cooperation between seller and buyer. We assume that unit marketing expenditure and unit price charged by the buyer influence the demand of the product being sold. The relationships between seller and buyer will be modeled by non-cooperative and cooperative games, respectively. The non-cooperative game is based on the Stackelberg strategy solution concept, where we consider separately the case when the seller is the leader (Seller-Stackelberg) and also when the buyer is the leader (Buyer-Stackelberg). Pareto efficient solutions will be provided for the cooperative game model. Numerical examples presented in this paper, including sensitivity analysis of some key parameters, will compare the results between different models considered.     

Using a partially observable Markov chain model to assess colonoscopy screening strategies – A cohort study

Colorectal cancer (CRC) is notoriously hard to combat for its high incidence and mortality rates. However, with improved screening technology and better understanding of disease pathways, CRC is more likely to be detected at early stage and thus more likely to be cured. Among the available screening methods, colonoscopy is most commonly used in the U.S. because of its capability of visualizing the entire colon and removing the polyps it detected. The current national guideline for colonoscopy screening recommends an observation-based screening strategy. Nevertheless, there is scant research studying the cost-effectiveness of the recommended observation-based strategy and its variants. In this paper, we describe a partially observable Markov chain (POMC) model which allows us to assess the cost-effectiveness of both fixed-interval and observation-based colonoscopy screening strategies. In our model, we consider detailed adenomatous polyp states and estimate state transition probabilities based on longitudinal clinical data from a specific population cohort. We conduct a comprehensive numerical study which investigates several key factors in screening strategy design, including screening frequency, initial screening age, screening end age, and screening compliance rate. We also conduct sensitivity analyses on the cost and quality of life parameters. Our numerical result demonstrates the usability of our model in assessing colonoscopy screening strategies with consideration of partial observation of true health states. This research facilitates future design of better colonoscopy screening strategies.     

Hybrid defensive resource allocations in the face of partially strategic attackers in a sequential defender–attacker game

Many models have been developed to study homeland security games between governments (defender) and terrorists (attacker, adversary, enemy), with the limiting assumption of the terrorists being rational or strategic. In this paper, we develop a novel hybrid model in which a centralized government allocates defensive resources among multiple potential targets to minimize total expected loss, in the face of a terrorist being either strategic or non-strategic. The attack probabilities of a strategic terrorist are endogenously determined in the model, while the attack probabilities of a non-strategic terrorist are exogenously provided. We study the robustness of defensive resource allocations by comparing the government’s total expected losses when: (a) the government knows the probability that the terrorist is strategic; (b) the government falsely believes that the terrorist is fully strategic, when the terrorist could be non-strategic; and (c) the government falsely believes that the terrorist is fully non-strategic, when the terrorist could be strategic. Besides providing six theorems to highlight the general results, we find that game models are generally preferred to non-game model even when the probability of a non-strategic terrorist is significantly greater than 50%. We conclude that defensive resource allocations based on game-theoretic models would not incur too much additional expected loss and thus more preferred, as compared to non-game-theoretic models.     

A steady Navier–Stokes model for compressible fluid with partially strong solutions

In this Note, we prove the existence of a partially strong solution to the steady Navier–Stokes equations for viscous barotropic compressible fluids, in a bounded simply connected domain of ${\mathbb{R}}^3$ with the prescribed generalized impermeability conditions ${\mathbf{curl}}^k\mathbf{u}?\mathbf{n}=0$, $k=0,1,2$ on the boundary. We call the solution “partially strong” because only the divergence-free part of the velocity field and the associated effective pressure have regularity typical for strong solution, while the density and the gradient part of the velocity have regularity typical for weak solution.     

A mean–variance optimization problem for discounted Markov decision processes

In this paper, we consider a mean–variance optimization problem for Markov decision processes (MDPs) over the set of (deterministic stationary) policies. Different from the usual formulation in MDPs, we aim to obtain the mean–variance optimal policy that minimizes the variance over a set of all policies with a given expected reward. For continuous-time MDPs with the discounted criterion and finite-state and action spaces, we prove that the mean–variance optimization problem can be transformed to an equivalent discounted optimization problem using the conditional expectation and Markov properties. Then, we show that a mean–variance optimal policy and the efficient frontier can be obtained by policy iteration methods with a finite number of iterations. We also address related issues such as a mutual fund theorem and illustrate our results with an example.     

A Note on Wiener–Hopf Factorization for Markov Additive Processes

We prove the Wiener–Hopf factorization for Markov additive processes. We derive also Spitzer–Rogozin theorem for this class of processes which serves for obtaining Kendall’s formula and Fristedt representation of the cumulant matrix of the ladder epoch process. Finally, we also obtain the so-called ballot theorem.     

Filltering of a partially observed process in the case of a high signal –to–noise ratio for correlated systems

This paper concerns a nonlinear filtering problem with correlated noises in the case of a high signal–to–noise ratio, when only one component of the signal is observed. We compute an approximate filter for the unnormalized filter associated to the system and derive both a Zakai and a Kushner-Stratonovitch type equation for the approximate filter     

A natural order in dynamical systems based on Conley–Markov matrices

We introduce a natural order to study properties of dynamical systems, especially their invariant sets. The new concept is based on the classical Conley index theory and transition probabilities among neighborhoods of different invariant sets when the dynamical systems are perturbed by white noises. The transition probabilities can be determined by the Fokker–Planck equation and they form a matrix called a Markov matrix. In the limiting case when the random perturbation is reduced to zero, the Markov matrix recovers the information given by the Conley connection matrix. The Markov matrix also produces a natural order from the least to the most stable invariant sets for general dynamical systems. In particular, it gives the order among the local extreme points if the dynamical system is a gradient-like flow of an energy functional. Consequently, the natural order can be used to determine the global minima for gradient-like systems. Some numerical examples are given to illustrate the Markov matrix and its properties.     

Co-op advertising models in manufacturer–retailer supply chains: A game theory approach

In the literature of cooperative (co-op) advertising, the focus of research is on a relationship in which a manufacturer is the leader and retailers are followers. This relationship implies the dominance of the manufacturer over retailers. Recent market structure reviews have shown a shift of retailing power from manufacturers to retailers. Retailers have equal or even greater power than a manufacturer when it comes to retailing. Based on this new market phenomenon, we intend to explore the role of vertical co-op advertising efficiency with respect to transactions between a manufacturer and a retailer through brand name investments, local advertising expenditures, and sharing rules of advertising expenses. Three co-op advertising models are discussed which are based on two noncooperative games and one cooperative game. In a leader–follower noncooperative game, the manufacturer is assumed to be a leader who first specifies the brand name investment and the co-op subsidization policy. The retailer, as a follower, then decides on the local advertising level. In a noncooperative simultaneous move game, the manufacturer and the retailer are assumed to act simultaneously and independently. In a cooperative game, the system profit is maximized for every Pareto efficient co-op advertising scheme, but not for any other schemes. All Pareto efficient co-op advertising schemes are associated with a single local advertising level and a single brand name investment level, but with variable sharing policies of advertising expenses. The best Pareto efficient advertising scheme is obtained taking members' risk attitudes into account. Utilizing the Nash bargaining model, we discuss two situations that (a) both members are risk averse, and (b) both members are risk neutral. Our results are consistent with the bargaining literature.     

A Large Deviation Principle for Symmetric Markov Processes with Feynman–Kac Functional

We establish a large deviation principle for the occupation distribution of a symmetric Markov process with Feynman–Kac functional. As an application, we show the L ^p -independence of the spectral bounds of a Feynman–Kac semigroup. In particular, we consider one-dimensional diffusion processes and show that if no boundaries are natural in Feller’s boundary classification, the L ^p -independence holds, and if one of the boundaries is natural, the L ^p -independence holds if and only if the L ²-spectral bound is non-positive.     

A note on chaotic and predictable representations for Itô–Markov additive processes

In this article, we provide predictable and chaotic representations for Itô–Markov additive processes X. Such a process is governed by a finite-state continuous time Markov chain J which allows one to modify the parameters of the Itô-jump process (in so-called regime switching manner). In addition, the transition of J triggers the jump of X distributed depending on the states of J just prior to the transition. This family of processes includes Markov modulated Itô–Lévy processes and Markov additive processes. The derived chaotic representation of a square-integrable random variable is given as a sum of stochastic integrals with respect to some explicitly constructed orthogonal martingales. We identify the predictable representation of a square-integrable martingale as a sum of stochastic integrals of predictable processes with respect to Brownian motion and power-jumps martingales related to all the jumps appearing in the model. This result generalizes the seminal result of Jacod–Yor and is of importance in financial mathematics. The derived representation then allows one to enlarge the incomplete market by a series of power-jump assets and to price all market-derivatives.     

A maximum principle for Markov regime-switching forward–backward stochastic differential games and applications

In this paper, we present an optimal control problem for stochastic differential games under Markov regime-switching forward–backward stochastic differential equations with jumps. First, we prove a sufficient maximum principle for nonzero-sum stochastic differential games problems and obtain equilibrium point for such games. Second, we prove an equivalent maximum principle for nonzero-sum stochastic differential games. The zero-sum stochastic differential games equivalent maximum principle is then obtained as a corollary. We apply the obtained results to study a problem of robust utility maximization under a relative entropy penalty and to find optimal investment of an insurance firm under model uncertainty.     

A feedback nash equilibrium solution for non–cooperative innovations in a stochastic differential game framework

In this paper, a stochastic model of non–cooperative technological innovations is developed. A feedback Sash equilibrium solution is obtained and the equilibrium innovation strategies are derived in explicit form. Several interesting properties of the equilibrium strategies are observed. On the one hand, an increase in the degree of competition in the industry, in the discount rate or in the state of technologyreduces innovation efforts. On the other hand, an increases the rate of degradation of the state of technology due to obsolescence results in an increase in innovation investment. While an increase in uncertainty reduces the expected present value of present and future discounted profits innovation efforts increase as uncertainty increases     

A least squares estimator for discretely observed Ornstein–Uhlenbeck processes driven by symmetric α-stable motions

We study the problem of parameter estimation for Ornstein–Uhlenbeck processes driven by symmetric α-stable motions, based on discrete observations. A least squares estimator is obtained by minimizing a contrast function based on the integral form of the process. Let h be the length of time interval between two consecutive observations. For both the case of fixed h and that of h → 0, consistencies and asymptotic distributions of the estimator are derived. Moreover, for both of the cases of h, the estimator has a higher order of convergence for the Ornstein–Uhlenbeck process driven by non-Gaussian α-stable motions (0 < α < 2) than for the process driven by the classical Gaussian case (α = 2).