Multilevel Stackelberg strategies in linear-quadratic systems

Open-loop multilevel Stackelberg strategies in deterministic, sequential decision-making problems for continuous linear systems and quadratic criteria are developed. Characterization of the Stackelberg controls via the solution of a higher-order square-matrix-Riccati differential equation is established; also, the basic structural properties of the coefficient matrices of this differential equation are established, and the basic structural properties of its solution are inferred.This work was supported in part by the Energy Research and Development Administration, Contract No. ERDA E(49-18)-2088.on leave from the Mihailo Pupin Institute, Belgrade, Yugoslavia.     

Stackelberg strategy solution for optimal software release policies

We present a software release policy which is based on the Stackelberg strategy solution concept. The model formulated assumes the existence of two type of producers in the market, the leader and follower. The resulting release policy combines both cost factors and a loss of opportunity factor which is the result of competition between the rival producers. We define a Stackelberg strategy pair in the context of our model and, through a series of preliminary results, show that an optimal strategy pair exists. We also present a numerical example which utilizes a software reliability growth model based on the nonhomogeneous Poisson process. Finally, we explore the relative leadership property of the optimal strategies.This work was supported in part by a FOAS Research Grant provided by RMIT. The author would like to thank the referees for constructive suggestions which helped to improve a previous version of this paper.     

Estimate of the Constant in Two Strengthened C.B.S. Inequalities for F.E.M. Systems of 2D Elasticity: Application to Multilevel Methods and a Posteriori Error Estimators

The constant γ in the strengthened Cauchy-Buniakowski-Schwarz (C.B.S.) inequality plays a crucial role in the convergence rate of multilevel iterative methods as well as in the efficiency of a posteriori error estimators, that is in the framework of finite element approximations of SPD problems. We consider the approximation of the 2D elasticity problem by the Courant element. Concerning multilevel convergence rate, that is the γ corresponding to nested general triangular meshes of size h and 2h, we have proved that γ²≤ 3/4$ uniformly on the mesh and the Poisson ratio. Concerning error estimator, that is the γ corresponding to quadratic and linear approximations on the same mesh, numerical computations have shown that the exact γ for a reference element deteriorates that is goes to one, when the Poisson ratio tends to 1/2     

Discrete storage processes and their Poisson flow and fluid flow approximations

Consider discrete storage processes that are modulated by environmental processes. Environmental processes cause interruptions in the input and/or output processes of the discrete storage processes. Due to the difficulties encountered in the exact analysis of such discrete storage systems, often Poisson flow and/or fluid flow models with the same modulating environmental processes are proposed as approximations for these systems. The analysis of Poisson flow and fluid flow models is much easier than that of the discrete storage processes. In this paper we give sufficient conditions under which the content of the discrete storage processes can be bounded by the Poisson flow and the fluid flow models. For example, we show that Poisson flow models and the fluid flow models developed by Kosten (and by Anick, Mitra and Sondhi) can be used to bound the performance of infinite (finite) source packetized voice/data communication systems. We also show that a Poisson flow model and the fluid flow model developed by Mitra can be used to bound the buffer content of a two stage automatic transfer line. The potential use of the bounding techniques presented in this paper, of course, transcends well beyond these examples.Supported in part by NSF grant DMS-9308149.     

A class of renewal Interrupted Poisson Processes and applications to queueing systems

Switched Poisson Processes and Interrupted Poisson Processes are often employed to characterize traffic streams in distributed computer and communications systems, especially in investigations of overflow processes in telecommunication networks. With these processes, input streams having inter-segment correlations and high variance as well as state-dependent traffic can properly be modelled. In this paper we first derive an approximation method to describe the Generalized Switched Poisson processes in conjunction with a renewal assumption. As a special case of this class of processes, the class of Interrupted Poisson processes is also included in the investigation. As a result, a generalization of the well-known class of Interrupted Poisson processes is obtained. It is shown that the renewal property is also given for this general class of Interrupted Poisson processes having generally distributed off-phase. To illustrate the accuracy of the presented renewal approximation of Generalized Switched Poisson processes and to show the major properties of the General Interrupted Poisson processes, applications to some basic queueing systems are discussed by means of numerical results.This work was done while the author was with Institute of Communications Switching and Data Technics, University of Stuttgart, Seidenstrasse 36, D-7000 Stuttgart 1, FRG.     

Characterization of Classical and Quantum Poisson Systems by Thinnings and Splittings

There exist several well–known characterizations of Poisson and mixed Poisson point processes (Cox processes) by thinning and splitting procedures. So a point process is necessarily a Cox process if for arbitrary small thinning parameter it can be obtained by a thinning of some other point process [30]. Poisson processes are characterized by the independence of the two random subconfigurations obtained by an independent splitting of the configuration into two parts [11]. For quantum mechanical particle systems beam splittings which are well–known in quantum optics provide analogous procedures. It is shown that coherent states respectively mixtures of them can be characterized in the same way as Poisson processes and Cox processes. Moreover, for the position distributions of these states which are “classical” point processes just the above mentioned characterizations are obtained. As example of mixed coherent states we consider Gaussian states which arise as equilibrium states of ideal Bose gases.     

Numerical simulations and modeling for stochastic biological systems with jumps

This paper gives a numerical method to simulate sample paths for stochastic differential equations (SDEs) driven by Poisson random measures. It provides us a new approach to simulate systems with jumps from a different angle. The driving Poisson random measures are assumed to be generated by stationary Poisson point processes instead of Lévy processes. Methods provided in this paper can be used to simulate SDEs with Lévy noise approximately. The simulation is divided into two parts: the part of jumping integration is based on definition without approximation while the continuous part is based on some classical approaches. Biological explanations for stochastic integrations with jumps are motivated by several numerical simulations. How to model biological systems with jumps is showed in this paper. Moreover, method of choosing integrands and stationary Poisson point processes in jumping integrations for biological models are obtained. In addition, results are illustrated through some examples and numerical simulations. For some examples, earthquake is chose as a jumping source which causes jumps on the size of biological population.     

Hierarchical mixture models for zero-inflated correlated count data

Count data with excess zeros are often encountered in many medical, biomedical and public health applications. In this paper, an extension of zero-inflated Poisson mixed regression models is presented for dealing with multilevel data set, referred as hierarchical mixture zero-inflated Poisson mixed regression models. A stochastic EM algorithm is developed for obtaining the ML estimates of interested parameters and a model comparison is also considered for comparing models with different latent classes through BIC criterion. An application to the analysis of count data from a Shanghai Adolescence Fitness Survey and a simulation study illustrate the usefulness and effectiveness of our methodologies.     

Dynamic feedback Stackelberg games with alternating leaders

In all past researches on dynamic Stackelberg games, the leader(s) and the followers are always assumed to be fixed. In practice, the roles of the players in a game may change from time to time. Some player in contract bridge, for example, acts as a leader at some stage but as a follower at the subsequent stage, which motivates the Stackelberg games with unfixed leaders. We aim to analyze the dynamic Stackelberg games with two players under such circumstances and call them dynamic Stackelberg games with alternating leaders. There are two goals in this paper. One goal is to establish models for a new type of games, dynamic Stackelberg games of alternating leaders with two players. The other goal is to extend dynamic programming algorithms to discrete time dynamic Stackelberg games with alternating leaders under feedback information structure.     

Optimum excess-loss reinsurance: a dynamic framework

The purpose of this article is to consider a two firms excess-loss reinsurance problem. The first firm is defined as the direct underwriter while the second firm is the reinsurer. As in the classical model of collective risk theory it is assumed that premium payments are received deterministically from policyholders at a constant rate, while the claim process is determined by a compound Poisson process. The objective of the underwriter is to maximize the expected present value of the long run terminal wealth (investments plus cash) of the firm by selecting an appropriate excess-loss coverage strategy, while the reinsurer seeks to maximize its total expected discounted profit by selecting an optimal loading factor. Since both firms' policies are interdependent we define an insurance game, solved by employing a Stackelberg solution concept. A diffusion approximation is used in order to obtain tractable results for a general claim size distribution. Finally, an example is presented illustrating computational procedures.     

Remarks on Multilevel Bases for Divergence-Free Finite Elements

The connection between the multilevel factorization method recently proposed by Sarin and Sameh for solving mixed discretizations of the Stokes equation using a divergence-free finite element formulation, and hierarchical basis preconditioners for the Poisson problem is established. For the 2D triangular Taylor–Hood element, a preconditioner is proposed that could be useful in fractional step methods.     

Normal convergence using Malliavin calculus with applications and examples

We prove the chain rule in the more general framework of the Wiener–Poisson space, allowing us to obtain the so-called Nourdin–Peccati bound. From this bound, we obtain a second-order Poincaré-type inequality that is useful in terms of computations. For completeness we survey these results on the Wiener space, the Poisson space, and the Wiener–Poisson space. We also give several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated fields (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein–Uhlenbeck SDE driven by fractional Brownian motion), Poisson functionals in the first Poisson chaos restricted to infinitely many “small” jumps (particularly fractional Lévy processes), and the product of two Ornstein–Uhlenbeck processes (one in the Wiener space and the other in the Poisson space). We also obtain bounds for their rate of convergence to normality.     

On martingale characterizations for some generalized space fractional Poisson processes

We obtain martingale characterizations for the generalized space fractional Poisson process (GSFPP) and for counting processes with Bern?tein intertimes. These serve as extensions of the Watanabe's characterization for the classical homogenous Poisson process. The corresponding assertion for the space fractional Poisson process (SFPP) is obtained as a particular case of our results.     

A note on the exponentiality of total hazards before failure

It is well known that a univariate counting process with a given intensity function becomes Poisson, with unit parameter, if the original time parameter is replaced by the integrated intensity. P. A. Meyer (in Martingales (H. Dinges, Ed.), pp. 32–37. Lecture Notes in Mathematics, Vol. 190, Springer-Verlag, Berlin) showed that a similar result holds for multivariate counting processes which have continuous compensators. Even more is true in the multivariate case: If each coordinate process is transformed individually according to a convenient time change, the resulting Poisson processes become independent. Our aim is to show that the continuity assumption of the compensators can be relaxed and, when the jumps of the compensator become small, we obtain the independent Poisson processes as a limit. An application for testing goodness-of-fit in survival analysis is given.     

On compound Poisson processes arising in change-point type statistical models as limiting likelihood ratios

Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In a previous paper of one of the authors it was established that one of these likelihood ratios, which is an exponential functional of a two-sided Poisson process driven by some parameter, can be approximated (for sufficiently small values of the parameter) by another one, which is an exponential functional of a two-sided Brownian motion. In this paper we consider yet another likelihood ratio, which is the exponent of a two-sided compound Poisson process driven by some parameter. We establish, that similarly to the Poisson type one, the compound Poisson type likelihood ratio can be approximated by the Brownian type one for sufficiently small values of the parameter. We equally discuss the asymptotics for large values of the parameter and illustrate the results by numerical simulations.     

Uniform Laws of Large Numbers for Triangular Arrays of Function-indexed Processes under Random Entropy Conditions

Mean Glivenko Cantelli Theorems are established for triangular arrays of rowwise independent processes. Methods developed by Pollard (1990) are combined with a truncation method essentially due to Alexander (1987). By this, applicability to partial sum processes in particular is achieved, for which Pollard’s truncation method fails. Nevertheless, the metric entropy condition appearing here is kept as weak as Pollard’s by means of application of Hoffmann-Jørgensen’s inequality, which has not been used so far in this context. The main theorem of the paper contains Pollard’s theorem as well as former results by Giné and Zinn (1984) and proves applicable to so-called random measure processes, certain function-indexed processes including empirical processes, partial-sum processes, the sequential empirical process and certain types of smoothed empirical processes. Statistical applications include nonparametric regression and the estimation of the intensity measure of a spatial Poisson process (Poisson point process).     

Multilevel Monte Carlo simulation for Lévy processes based on the Wiener–Hopf factorisation

In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was introduced for a large family of Lévy processes that is based on the Wiener–Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. Moreover, we provide here for the first time a theoretical analysis of the new Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its multilevel variant for computing expectations of functions depending on the historical trajectory of a Lévy process. We derive rates of convergence for both methods and show that they are uniform with respect to the “jump activity” (e.g. characterised by the Blumenthal–Getoor index). We also present a modified version of the algorithm in Kuznetsov et al. (2011) which combined with the multilevel methodology obtains the optimal rate of convergence for general Lévy processes and Lipschitz functionals. This final result is only a theoretical one at present, since it requires independent sampling from a triple of distributions which is currently only possible for a limited number of processes.     

Networks of infinite-server queues with nonstationary Poisson input

In this paper we focus on networks of infinite-server queues with nonhomogeneous Poisson arrival processes. We start by introducing a more general Poisson-arrival-location model (PALM) in which arrivals move independently through a general state space according to a location stochastic process after arriving according to a nonhomogeneous Poisson process. The usual open network of infinite-server queues, which is also known as a linear population process or a linear stochastic compartmental model, arises in the special case of a finite state space. The mathematical foundation is a Poisson-random-measure representation, which can be obtained by stochastic integration. It implies a time-dependent product-form result: For appropriate initial conditions, the queue lengths (numbers of customers in disjoint subsets of the state space) at any time are independent Poisson random variables. Even though there is no dependence among the queue lengths at each time, there is important dependence among the queue lengths at different times. We show that the joint distribution is multivariate Poisson, and calculate the covariances. A unified framework for constructing stochastic processes of interest is provided by stochastically integrating various functionals of the location process with respect to the Poisson arrival process. We use this approach to study the flows in the queueing network; e.g., we show that the aggregate arrival and departure processes at a given queue (to and from other queues as well as outside the network) are generalized Poisson processes (without necessarily having a rate or unit jumps) if and only if no customer can visit that queue more than once. We also characterize the aggregate arrival and departure processes when customers can visit the queues more frequently. In addition to obtaining structural results, we use the stochastic integrals to obtain explicit expressions for time-dependent means and covariances. We do this in two ways. First, we decompose the entire network into a superposition of independent networks with fixed deterministic routes. Second, we make Markov assumptions, initially for the evolution of the routes and finally for the entire location process. For Markov routing among the queues, the aggregate arrival rates are obtained as the solution to a system of input equations, which have a unique solution under appropriate qualifications, but not in general. Linear ordinary differential equations characterize the time-dependent means and covariances in the totally Markovian case.     

Stochastic averaging principles for multi-valued stochastic differential equations driven by poisson point Processes

The purpose of this article is to investigate an averaging principle for multi-valued stochastic differential equations (MSDEs) driven by Poisson point processes. The solutions to MSDEs driven by Poisson point processes can be approximated by solutions to averaged MSDEs in the sense of both convergence in mean square and convergence in probability. Finally, an example is presented to illustrate the averaging principle.