Joint modelling of the total amount and the number of claims by conditionals

In the risk theory context, let us consider the classical collective model. The aim of this paper is to obtain a flexible bivariate joint distribution for modelling the couple (S,N), where N is a count variable and S=X₁+?+X_N is the total claim amount. A generalization of the classical hierarchical model, where now we assume that the conditional distributions of S|N and N|S belong to some prescribed parametric families, is presented. A basic theorem of compatibility in conditional distributions of the type S given N and N given S is stated. Using a known theorem for exponential families and results from functional equations new models are obtained. We describe in detail the extension of two classical collective models, which now we call Poisson-Gamma and the Poisson-Binomial conditionals models. Other conditionals models are proposed, including the Poisson-Lognormal conditionals distribution, the Geometric-Gamma conditionals model and a model with inverse Gaussian conditionals. Further developments of collective risk modelling are given.     

Three-dimensional modelling of NOx and particulate traps using CFD: A porous medium approach

Lean burn after-treatment systems are the current focus for reducing emissions from diesel exhaust. The trend is for commercial CFD packages to use a single channel modelling approach. Due to computational demands, this necessitates specification of representative channels for modelling, implying prior knowledge of the flow field. This paper investigates a methodology for applying the porous medium approach to lean burn after-treatment systems. This approach has proved successful for three-way catalysis modelling and has the advantage that the flow field is predicted. Chemical kinetic rates for NO_x trapping and regeneration in the model are based on information available in the open literature. Similarly, filtration information based on mass accumulation and soot combustion kinetics are also readily available. Modification of the source terms in a commercial CFD package enables prediction of trapping and release of NO_x. This is an effective way to model a NO_x trap after-treatment system and provides simultaneous 3D modelling of the flow field. With diesel, particulate filtration is required. In the case of particulate traps, however, because of channel geometry, some assumptions are necessary for use of the porous medium approach and these are discussed in this paper. Both models produce qualitatively correct output and have parameters that can be tuned to conform to experimental data. Data to validate the NO_x trap model is to be measured. The particulate trap model, on the other hand, is a feasibility study for modelling the complete diesel after-treatment system using the porous medium approach.     

TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts

In this paper, we consider a portfolio of n dependent risks X₁,…,X_n and we study the stochastic behavior of the aggregate claim amount S=X₁+?+X_n. Our objective is to determine the amount of economic capital needed for the whole portfolio and to compute the amount of capital to be allocated to each risk X₁,…,X_n. To do so, we use a top-down approach. For (X₁,…,X_n), we consider risk models based on multivariate compound distributions defined with a multivariate counting distribution. We use the TVaR to evaluate the total capital requirement of the portfolio based on the distribution of S, and we use the TVaR-based capital allocation method to quantify the contribution of each risk. To simplify the presentation, the claim amounts are assumed to be continuously distributed. For multivariate compound distributions with continuous claim amounts, we provide general formulas for the cumulative distribution function of S, for the TVaR of S and the contribution to each risk. We obtain closed-form expressions for those quantities for multivariate compound distributions with gamma and mixed Erlang claim amounts. Finally, we treat in detail the multivariate compound Poisson distribution case. Numerical examples are provided in order to examine the impact of the dependence relation on the TVaR of S, the contribution to each risk of the portfolio, and the benefit of the aggregation of several risks.     

Modèles semi-factoriels et modèles de Néron

Let S be the spectrum of a discrete valuation ring with function field K. Let X be a scheme over S. We will say that X is semi-factorial over S if any invertible sheaf on the generic fiber X _K can be extended to an invertible sheaf on X. Here we show that any proper geometrically normal scheme over K admits a proper, flat, normal and semi-factorial model over S. We also construct some semi-factorial compactifications of regular S-schemes, such as Néron models of abelian varieties. The semi-factoriality property for a scheme X/S corresponds to the Néron property of its Picard functor. In particular, one can recover the Néron model of the Picard variety ${{\rm Pic}_{X_K/K,{\rm red}}^0}$ of X _K from the Picard functor Pic _X/S , as in the known case of curves. This provides some information on the relative algebraic equivalence on the S-scheme X.     

New approaches in computational dynamics of the mixing flow

Between the most mature interdisciplinary areas, computational fluid dynamics (CFD) comes recently into focus. In the same time, it becomes more and more difficult to contribute fundamental research to it. However, although it remains unpredictable how CFD develops, it is part of what makes it an exciting and attractive discipline.This paper aims to exhibit some part of recent work in CFD. It concerns the qualitative approach of the turbulent behaviour of a mixing flow in an excitable media. Studying a mixing for a flow implies the analysis of successive stretching and folding phenomena for its particles, the influence of parameters and initial conditions. In the previous works, the study of the 3D non-periodic models exhibited a quite complicated behaviour. In agreement with experiments, there were involved some significant events, the so-called “rare events”. The variation of parameters had a great influence on the length and surface deformations. The experiments were realized with a special vortex installation, it was used a well-known aquatic algae as biologic material, and the water as basic fluid.In the paper there are presented some features of a qualitative comparative analysis of the model associated to the vortex flow technology. In the computational analysis there were used the fast analysis tools of MAPLE11 soft, in order to check the “rare events” and to complete the statistical analysis of the behaviour of the model.     

A simple characterization of Student’s distributions and normal distributions

Let X_1:3≤X_2:3≤X_3:3 be the order statistics of the independent and identically distributed random variables X₁, X₂, and X₃. Using the regressional property E[(aX_2:3−X_1:3)²|X_2:3=x]=E[(X_3:3−aX_2:3)²|X_2:3=x], we characterize some distributions, including Student’s t_ν distributions (ν>2) and normal distributions.     

The jumping phenomenon of the dimensions of cohomology groups of tangent sheaf

Let X be a compact complex manifold. Consider a small deformation π: X → B of X, the dimensions of the cohomology groups of tangent sheaf H^q(X_t, TX_t) may vary under this deformation. This article studies such phenomena by studying the obstructions to deform a class in H^q(X, TX) with parameter t and gets a formula for the obstructions.     

Derivations, the Lawrence–Sullivan interval and the Fiorenza–Manetti mapping cone

We describe the rational homotopy type of any component of the based mapping space map^*(X,Y) as an explicit L _∞ algebra defined on the (desuspended and positive) derivations between Quillen models of X and Y. When considering the Lawrence–Sullivan model of the interval, we obtain an L _∞ model of the contractible path space of Y. We then relate this, in a geometrical and natural manner, to the L _∞ structure on the Fiorenza–Manetti mapping cone of any differential graded Lie algebra morphism, two in principal different algebraic objects in which Bernoulli numbers appear.     

Extremes of conditioned elliptical random vectors

Let {X_n,n?1} be iid elliptical random vectors in R^d,d≥2 and let I,J be two non-empty disjoint index sets. Denote by X_n,I,X_n,J the subvectors of X_n with indices in I,J, respectively. For any a∈R^d such that a_J is in the support of X_1,J the conditional random sample X_n,I|X_n,J=a_J,n≥1 consists of elliptically distributed random vectors. In this paper we investigate the relation between the asymptotic behaviour of the multivariate extremes of the conditional sample and the unconditional one. We show that the asymptotic behaviour of the multivariate extremes of both samples is the same, provided that the associated random radius of X₁ has distribution function in the max-domain of attraction of a univariate extreme value distribution.     

Distribution-free inference in record series

Let {X _t , t = 1} be a time series. A (upper) record is a value X _j such that X _j > max{X ₁,…,X _j-1}. Some popular models in record theory are the Yang-Nevzorov and the Linear Drift models. The stochastic behavior of records under these models has been much studied and many interesting distribution-free properties have been unearthed. However the estimation of the parameters of these models has been less explored. This work introduces some estimators of these parameters. Their behavior is investigated theoretically and by numerical simulations. It is shown that some of these estimators are easy to compute and their asymptotic properties are accurate and distribution-free. Some goodness-of-fit tests for these models are also presented.     

Dynamic Markov bridges motivated by models of insider trading

Given a Markovian Brownian martingale Z, we build a process X which is a martingale in its own filtration and satisfies X₁=Z₁. We call X a dynamic bridge, because its terminal value Z₁ is not known in advance. We compute its semimartingale decomposition explicitly under both its own filtration F^X and the filtration F^X,Z jointly generated by X and Z. Our construction is heavily based on parabolic partial differential equations and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading that can be viewed as a non-Gaussian generalization of the model of Back and Pedersen (1998) [3], where the insider’s additional information evolves over time.     

Lévy random bridges and the modelling of financial information

The information-based asset-pricing framework of Brody-Hughston-Macrina (BHM) is extended to include a wider class of models for market information. To model the information flow, we introduce a class of processes called Lévy random bridges (LRBs), generalising the Brownian bridge and gamma bridge information processes of BHM. Given its terminal value at T, an LRB has the law of a Lévy bridge. We consider an asset that generates a cash-flow X_T at T. The information about X_T is modelled by an LRB with terminal value X_T. The price process of the asset is worked out, along with the prices of options.     

Empirical Mark Covariance and Product Density Function of Stationary Marked Point Processes—A Survey on Asymptotic Results

Marked point processes are stochastic models to describe random patterns of marked points {[X _i ,M _i ], i?≥?1} in some bounded subset of the d-dimensional Euclidean space (usually d?=?1, 2 or 3 in applications), where each point X _i carries additional random information expressed as mark M _i taking values in some metric space. To study the correlations between distinct points and between marks located at distinct points we use kernel-type estimators of the second-order product density and the mark covariance function of a spatially homogeneous marked point process. Both functions and their empirical counterparts are suitable characteristics to identify point process models by construction of statistical goodness-of-fit tests.     

Time-dependent analysis of virtual waiting time behaviour in discrete time queues

Discrete time queueing models have been shown previously to be of practical use for modelling the approximate time-dependent behaviour of queue length in systems of the form M(t)/G/c. In this paper we extend these models to include the time-dependent behaviour of virtual waiting time.     

On the Stability of One Characterization of Stable Distributions

In 1923, G. Polya proved that if X ₁ and X ₂ are independent identically distributed random variables (i.i.d.r.v.) with finite variance, then the distributions of X ₁ and (X ₁+X ₂)/ $\sqrt 2$ are coincidental iff X ₁ has the normal distribution with zero mean. Is an analogous theorem possible for an couple of statistics X ₁ and (X ₁+X ₂)/2^1/α if α<2? P. Lévy constructed an example that denies that hypothesis. However, having supplemented the condition of coincidence of the distributions of X ₁ and (X ₁+X ₂)/2^1/α with a similar condition, namely, requiring, in addition, for the distributions of X ₁ and (X ₁+X ₂+X ₃)/3^1/α to be coincident (here X ₁,X ₂ and X ₃ are i.i.d.r.v.), P. Lévy has proved that X ₁ and X ₂ have a strictly stable distribution. The stability of this characterization in a metric λ₀ (that is defined in the class of characteristic functions by analogy with a uniform metric defined in the class of distributions) without an additional symmetry assumption as well as the stability in a Lévy metric L are analizied in this paper.     

On plane models for Drinfeld modular curves

In this work, we find plane models for certain Drinfeld modular curves X₀(n) which have better properties than the plane models derived from the usual Drinfeld modular equations. As an application, we construct ring class fields over imaginary quadratic fields by using singular values of generators of the function field of X₀(n).     

Solving nonconvex nonlinear programming problems via a new aggregate constraint homotopy method

In this paper, by introducing C² mappings ξ_i(x,z_i),i=1,…,m and using the idea of the aggregate function method, a new aggregate constraint homotopy method is proposed to solve the Karush-Kuhn-Tucker (KKT) point of nonconvex nonlinear programming problems. Compared with the previous results, the choice scope of initial points is greatly enlarged, so use of the new aggregate constraint homotopy method may improve the computational efficiency of reduced predictor-corrector algorithms.     

Nonlinear analysis of the orbital motions of immersed rotors using a spectral/Galerkin approach

We recently developed a symbolic-numerical formulation for the nonlinear planar motion of rotors under fluid confinement, based on a spectral/Galerkin approach, for gap geometries of about δ=H/R≈0.1––where H is the average annular gap and R is the rotor radius. Results showed a quite good agreement between the class of approximate models generated, the corresponding analytical exact planar model and experiments. This methodology can be almost entirely automated on a symbolic computing environment.In the present paper this symbolic-numerical spectral/Galerkin procedure is extended in order to deal with nonlinear orbital motions––X(t) and Y(t) taking place in orthogonal directions.Numerical simulations performed over a centered rotor configuration maintained by nonisotropic supports (K^st_Y/K^st_X=0.4, where K^st_X and K^st_Y stand for the structural stiffnesses), which exhibit interesting dynamics, show a quite good agreement between this type of approximate models and the corresponding analytical exact (but quite involved) model, developed in the past by the authors.With the proposed symbolic-numerical approach one can obtain accurate nonlinear dynamical formulations enabling the study, understanding and prediction of nonlinear orbital rotor dynamics.     

Identification of parameters by the distribution of the minimum: The tri-variate normal case with negative correlations

Let (X₁,X₂,X₃) be a 3-variate normal vector with zero means and a non-singular co-variance matrix Σ, where for i≠j, Σ_ij≤0. It is shown here that it is then possible to determine the three variances and the three correlations based only on the knowledge of the density of the minimum {X₁,X₂,X₃}. Our method consists of careful determination and analysis of the asymptotic orders of various bivariate tail probabilities.     

A Local Refinement Strategy for Constructive Quantization of Scalar SDEs

We present a fully constructive method for quantization of the solution X of a scalar SDE in the path space L _p [0,1] or C[0,1]. The construction relies on a refinement strategy which takes into account the local regularity of X and uses Brownian motion (bridge) quantization as a building block. Our algorithm is easy to implement, its computational cost is close to the size of the quantization, and it achieves strong asymptotic optimality provided this property holds for the Brownian motion (bridge) quantization.