Poisson cohomology of broken Lefschetz fibrations

We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology around singular points we adapt techniques developed for the Sklyanin algebra. As a side result, we give compact formulas for the Poisson coboundary operator of an arbitrary Jacobian Poisson structure in 4 dimensions.     

Critical phenomenon for a percolation model

We consider a percolation model which consists of oriented lines placed randomly on the plane. The lines are of random length and at a random angle with respect to the horizontal axis and are placed according to a Poisson point process; the length, angle, and orientation being independent of the underlying Poisson process. We establish a critical behaviour of this model, i.e., percolation occurs for large intensity of the Poisson process and does not occur for smaller intensities. In the special case when the lines are of fixed unit length and are either oriented vertically up or oriented horizontally to the left, with probability p or (1-p), respectively, we obtain a lower bound on the critical intensity of percolation.     

Vector Bundle-Valued Poisson and Cauchy Kernel Functions on Classical Domains

In this paper we investigate the vector bundle-valued Cauchy and Poisson kernel functions. We compute explicitly matrix-valued eigenfunctions of an invariant differential operator on the classical domain of Type I. Furthermore, a special choice of a vector bundle gives us a matrix-valued Cauchy and Poisson kernel function which satisfies the matrix-valued Laplacian operator on the classical domain of Type I.     

Fragmentation associated with Lévy processes using snake

We consider the height process of a Lévy process with no negative jumps, and its associated continuous tree representation. Using Lévy snake tools developed by Le Gall-Le Jan and Duquesne-Le Gall, with an underlying Poisson process, we construct a fragmentation process, which in the stable case corresponds to the self-similar fragmentation described by Miermont. For the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment. We also give a special Markov property for the snake which is of its own interest.      

Approximate analysis of exponential tandem queues with blocking

In this paper, an approximate method for the analysis of open networks of queues in tandem and with blocking is proposed. The network consists of M single server queuing stations with exogenous Poisson arrival processes and exponentially distributed service times. The analysis is based on the method of decomposition where the total network is broken down into queues which are analyzed as M/C₂/1/N queues assuming Poisson arrival and departure processes to find the steady-state probabilities of the number of customers at each station. The procudure reduces the problem to a number of elementary operations which can be performed efficiently with the aid of a computer. We also compare different definitions of blocking. Numerical results are given to demonstrate the accuracy of the new method.     

Non-homogeneous space-time fractional Poisson processes

The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto (2012), is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson process (SFPP). We study the fractional generalization of the non-homogeneous Poisson process and call it the non-homogeneous space-time fractional Poisson process (NHSTFPP). We compute their pmf and generating function and investigate the associated differential equation. The limit theorems for the NHSTFPP process are studied. We study the distributional properties, the asymptotic expansion of the correlation function of the non-homogeneous time fractional Poisson process (NHTFPP) and subsequently investigate the long-range dependence (LRD) property of a special NHTFPP. We investigate the limit theorem for the fractional non-homogeneous Poisson process (FNHPP) studied by Leonenko et al. (2014). Finally, we present some simulated sample paths of the NHSTFPP process.     

Algorithms to Find Exact Inclusion Probabilities for Conditional Poisson Sampling and Pareto πps Sampling Designs

Conditional Poisson Sampling Design as developed by Haje´k may be defined as a Poisson sampling conditioned by the requirement that the sample has fixed size. In this paper, an algorithm is implemented to calculate the conditional inclusion probabilities given the inclusion probabilities under Poisson Sampling. A simple algorithm is also given for second order inclusion probabilities in Conditional Poisson Sampling. Furthermore a numerical method is introduced to compute the unconditional inclusion probabilities when the conditional inclusion probabilities are predetermined. Simultaneously, we study the Pareto ps sampling design. This method, introduced by Rose´n, belongs to a class of sampling schemes called Order Sampling with Fixed Distribution Shape. Methods are provided to compute the first and second order inclusion probabilities numerically also in this case, as well as two procedures to adjust the parameters to get predetermined inclusion probabilities.     

Asymptotic distribution of the downtime of a monotone system

To compute the distribution of the downtime in a time interval [0,t] for a highly available monotone system, a Compound Poisson process (CP(t)) approximation is often used. In this paper we give sufficient conditions for when the distribution ofCP(t) is an asymptotic limit. We also study the convergence to the normal distribution.     

Spectrally negative Lévy processes with Parisian reflection below and classical reflection above

We consider a company that receives capital injections so as to avoid ruin. Differently from the classical bail-out settings, where the underlying process is restricted to stay at or above zero, we study the case bail-out can only be made at independent Poisson observation times. Namely, we study a version of the reflected process that is pushed up to zero only on Poisson arrival times at which the process is below zero. We also study the case with additional classical reflection above so as to model a company that pays dividends according to a barrier strategy. Focusing on the spectrally negative Lévy case, we compute, using the scale function, various fluctuation identities, including capital injections and dividends.     

Poisson cohomology in dimension two

It is known that the computation of the Poisson cohomology is closely related to the classification of singularities of Poisson structures. In this paper, we will first look for the normal forms of germs at (0,0) of Poisson structures onG ² (G=ℝ or ℂ) and recall a result given by Arnold. Then we will compute locally the Poisson cohomology of a particular type of Poisson structure.     

A Saddlepoint Approximation to the Distribution of Inhomogeneous Discounted Compound Poisson Processes

In this article we propose an accurate approximation to the distribution of the discounted total claim amount, where the individual claim amounts are independent and identically distributed and the number of claims over a specified period is governed by an inhomogeneous Poisson process. More precisely, we compute cumulant generating functions of such discounted total claim amounts under various intensity functions and individual claim amount distributions, and invert them by the saddlepoint approximation. We provide precise conditions under which the saddlepoint approximation holds. The resulting approximation is numerically accurate, computationally fast and hence more efficient than Monte Carlo simulation.     

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In this paper, we extend the previous Markov-modulated reflected Brownian motion model discussed in [1] to a Markov-modulated reflected jump diffusion process, where the jump component is described as a Markov-modulated compound Poisson process. We compute the joint stationary distribution of the bivariate Markov jump process. An abstract example with two states is given to illustrate how the stationary equation described as a system of ordinary integro-differential equations is solved by choosing appropriate boundary conditions. As a special case, we also give the sationary distribution for this Markov jump process but without Markovian regime-switching.     

On the Ruin Problem in a Markov-Modulated Risk Model

In this paper, we consider the compound Poisson risk model influenced by an external Markovian environment process, i.e. Markov-modulated compound Poisson model. The explicit Laplace transforms of Gerber–Shiu functions are obtained, while the explicit Gerber–Shiu functions are derived for the K _n -family claim size distributions in the two-states case.      

Curved Koszul duality of algebras over unital versions of binary operads

We develop a curved Koszul duality theory for algebras presented by quadratic-linear-constant relations over unital versions of binary quadratic operads. As an application, we study Poisson n-algebras given by polynomial functions on a standard shifted symplectic space. We compute explicit resolutions of these algebras using curved Koszul duality. We use these resolutions to compute derived enveloping algebras and factorization homology on parallelized simply connected closed manifolds with coefficients in these Poisson n-algebras.     

Modelling and analysis of a software system with improvements in the testing phase

In this paper, we consider the error detection phenomena in the testing phase when modifications or improvements can be made to the software in the testing phase. The occurrence of improvements is described by a homogeneous Poisson process with intensity rate denoted by λ. The error detection phenomena is assumed to follow a nonhomogeneous Poisson process (NHPP) with the mean value function being denoted by m(t). Two models are presented and in one of the models, we have discussed an optimal release policy for the software taking into account the occurrences of errors and improvements. Finally, we discuss the possibility of an improvement removing k errors with probability p_k, k ≥ 0 in the software and develop a NHPP model for the error detection phenomena in this situation.     

Performance appraisal based on distance function methods

Performance appraisal is a process used by some firms to evaluate their employees’ efficiency and productivity in order to plan their promotion policy, salary policy, layoffs policy, etc. Initially this process was just carried out by the executive staff, but recently it has evolved into an evaluation process based on the opinion of different reviewers, supervisors, collaborators, customers and the employees themselves (360-degree method). In such an evaluation process the reviewers evaluate some indicators related to employees performance appraisal. In this paper we propose an evaluation framework where there are different sets of reviewers taking part in the evaluation process. Since reviewers have a different knowledge about the evaluated employee, it seems suitable to offer a flexible framework in which different reviewers can express their assessments in different finite scales according to their knowledge. The final aim is to compute a global evaluation for each employee, that can be used by the management team to make their decisions regarding their human resources policy. In this way, to obtain a global evaluation for each employee, we propose a methodology able to aggregate individual valuation in a metric L_p framework. In this context, the associated optimization problems can be reduced to an Extended Goal Programming formulation that is very easy to compute.     

The variational Poisson cohomology

It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix.     

White Noise of Poisson Random Measures

We develop a white noise theory for Poisson random measures associated with a pure jump Lévy process. The starting point of this theory is the chaos expansion of Itô. We use this to construct the white noise of a Poisson random measure, which takes values in a certain distribution space. Then we show, how a Skorohod/Itô integral for point processes can be represented by a Bochner integral in terms of white noise of the random measure and a Wick product. Further, based on these concepts we derive a generalized Clark–Haussmann–Ocone theorem with respect to a combination of Gaussian noise and pure jump Lévy noise. We apply this theorem to obtain an explicit formula for partial observation minimal variance portfolios in financial markets, driven by Lévy processes. As an example we compute the closest hedge to a binary option.     

On the First Entry Time of a ℤ+-valued AR(1) Process

In this paper we derive an explicit formula for the expected value of the first time a ₊-valued AR(1) process exceeds a given level. Using martingale theory we obtain a generalized Wald's equation that holds under a simple integrability condition. As an application, we give an asymptotic formula for the expected value of the first exit time of the AR(1) process with a thinned Poisson innovation.     

Fractional Poisson process

A fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov–Feller equation. We have found the probability of n arrivals by time t for fractional stream of events. The fractional Poisson process captures long-memory effect which results in non-exponential waiting time distribution empirically observed in complex systems. In comparison with the standard Poisson process the developed model includes additional parameter μ. At μ=1 the fractional Poisson becomes the standard Poisson and we reproduce the well known results related to the standard Poisson process.As an application of developed fractional stochastic model we have introduced and elaborated fractional compound Poisson process.