Properties of Hida processes on . 2. Prediction and interpolation problems for processes on

If E is an ordered set, we study the processes Y_t, t E, for which the vectorial spaces _t generated by all the conditional expectations E(Y_sβ _t) for s ≥ t have finite dimensions d(t) ≤ N. ( _t is some convenient filtration.) We first develop a geometrical approach in the general situation and give a “Goursat's representation” Y_t = Σf_i(t)M_i(t), where the M_i(t) are martingales. We then restrict us to the cases E = or E = ² and give representations of the processes by the mean of stochastic integrals of “Goursat's kernels.” The special case when Y_t is the solution of a differential equation is considered.     

Representing probability measures using probabilistic processes

In the Type-2 Theory of Effectivity, one considers representations of topological spaces in which infinite words are used as “names” for the elements they represent. Given such a representation, we show that probabilistic processes on infinite words, under which each successive symbol is determined by a finite probabilistic choice, generate Borel probability measures on the represented space. Conversely, for several well-behaved types of space, every Borel probability measure is represented by a corresponding probabilistic process. Accordingly, we consider probabilistic processes as providing “probabilistic names” for Borel probability measures. We show that integration is computable with respect to the induced representation of measures.     

Quantum Lévy processes on dual groups

We present a theory of quantum (non-commutative) Lévy processes on dual groups which generalizes the theory of Lévy processes on bialgebras. It follows from a result of N. Muraki that there exist exactly 5 notions of non-commutative ‘positive’ stochastic independence. We show that one can associate a commutative bialgebra with each pair consisting of a dual group and one of the 5 notions of independence. This construction is related to a construction of U. Franz. Our construction has the advantage that the important case of free independence is included. We show that Lévy processes are given by their generators which are precisely the conditonally positive linear functionals on the dual group.Supported by the European Research Training Network “Quantum Probability with Applications to Physics, Information Theory and Biology”     

Self-intersection local time for Gaussian ′( )-processes: Existence, path continuity and examples

In this paper we develop a criterion for existence or non-existence of self-intersection local time (SILT) for a wide class of Gaussian ′( ^d)-valued processes, we show that quite generally the SILT process has continuous paths, and we give several examples which illustrate existence of SILT for different ranges of dimensions (e.g., d ≤ 3, d ≤ 7 and 5 ≤ d ≤ 11 in the Brownian case). Some of the examples involve branching and exhibit “dimension gaps”. Our results generalize the work of Adler and coauthors, who studied the special case of “density processes” and proved that SILT paths are cadlag in the Brownian case making use of a “particle picture” approximation (this technique is not available for our general formulation).     

Production/Clearing Models Under Continuous and Sporadic Reviews

We consider production/clearing models where random demand for a product is generated by customers (e.g., retailers) who arrive according to a compound Poisson process. The product is produced uniformly and continuously and added to the buffer to meet future demands. Allowing to operate the system without a clearing policy may result in high inventory holding costs. Thus, in order to minimize the average cost for the system we introduce two different clearing policies (continuous and sporadic review) and consider two different issuing policies (“all-or-some” and “all-or-none”) giving rise to four distinct production/clearing models. We use tools from level crossing theory and establish integral equations representing the stationary distribution of the buffer’s content level. We solve the integral equations to obtain the stationary distributions and develop the average cost objective functions involving holding, shortage and clearing costs for each model. We then compute the optimal value of the decision variables that minimize the objective functions. We present numerical examples for each of the four models and compare the behaviour of different solutions.AMS 2000 Subject Classification: 90B05 Inventory, storage, reservoirs; 90B22 Queues and service; 90B30 Production models     

A Generalized φ-Divergence for Asymptotically Multivariate Normal Models

I. Csiszár's (Magyar. Tud. Akad. Mat. Kutató Int. Közl8 (1963), 85–108) -divergence, which was considered independently by M. S. Ali and S. D. Silvey (J. R. Statist. Soc. Ser. B28 (1966), 131–142) gives a goodness-of-fit statistic for multinomial distributed data. We define a generalized φ-divergence that unifies the -divergence approach with that of C. R. Rao and S. K. Mitra (“Generalized Inverse of Matrices and Its Applications,” Wiley, New York, 1971) and derive weak convergence to a χ² distribution under the assumption of asymptotically multivariate normal distributed data vectors. As an example we discuss the application to the frequency count in Markov chains and thereby give a goodness-of-fit test for observations from dependent processes with finite memory.     

Mean residual life order of convolutions of heterogeneous exponential random variables

In this paper, we study convolutions of heterogeneous exponential random variables with respect to the mean residual life order. By introducing a new partial order (reciprocal majorization order), we prove that this order between two parameter vectors implies the mean residual life order between convolutions of two heterogeneous exponential samples. For the 2-dimensional case, it is shown that there exists a stronger equivalence. We discuss, in particular, the case when one convolution involves identically distributed variables, and show in this case that the mean residual life order is actually associated with the harmonic mean of parameters. Finally, we derive the “best gamma bounds” for the mean residual life function of any convolution of exponential distributions under this framework.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.     

Inhomogeneous Current States in a Gauged Two-Component Ginzburg-Landau Model

We consider the energy bounds of inhomogeneous current states in doped antiferromagnetic insulators in the framework of the two-component Ginzburg-Landau model. Using the formulation of this model in terms of the gauge-invariant order parameters (the unit vector n, spin stiffness field ρ², and particle momentum c), we show that this strongly correlated electron system involves a geometric small parameter that determines the degree of packing in the knots of filament manifolds of the order parameter distributions for the spin and charge degrees of freedom. We find that as the doping degree decreases, the filament density increases, resulting in a transition to an inhomogeneous current state with a free energy gain.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 182–189, July, 2005.     

Modelling Credit Risk in the Jump Threshold Framework

ABSTRACT

The jump threshold framework for credit risk modelling developed by Garreau and Kercheval enjoys the advantages of both structural- and reduced-form models. In their article, the focus is on multidimensional default dependence, under the assumptions that stock prices follow an exponential Lévy process (i.i.d. log returns) and that interest rates and stock volatility are constant. Explicit formulas for default time distributions and basket credit default swap (CDS) prices are obtained when the default threshold is deterministic, but only in terms of expectations when the default threshold is stochastic. In this article, we restrict attention to the one-dimensional, single-name case in order to obtain explicit closed-form solutions for the default time distribution when the default threshold, interest rate and volatility are all stochastic. When the interest rate and volatility processes are affine diffusions and the stochastic default threshold is properly chosen, we provide explicit formulas for the default time distribution, prices of defaultable bonds and CDS premia. The main idea is to make use of the Duffie–Pan–Singleton method of evaluating expectations of exponential integrals of affine diffusions.     

Polynomial growth rates

We consider linear equations v^′=A(t)v with a polynomial asymptotic behavior, that can be stable, unstable and central. We show that this behavior is exhibited by a large class of differential equations, by giving necessary and sufficient conditions in terms of generalized “polynomial” Lyapunov exponents for the existence of polynomial behavior. In particular, any linear equation in block form in a finite-dimensional space, with three blocks having “polynomial” Lyapunov exponents respectively negative, positive, and zero, has a nonuniform version of polynomial trichotomy, which corresponds to the usual notion of trichotomy but now with polynomial growth rates. We also obtain sharp bounds for the constants in the notion of polynomial trichotomy. In addition, we establish the persistence under sufficiently small nonlinear perturbations of the stability of a nonuniform polynomial contraction.     

The net Bayes premium with dependence between the risk profiles

In Bayesian analysis it is usual to assume that the risk profiles Θ₁ and Θ₂ associated with the random variables “number of claims” and “amount of a single claim”, respectively, are independent. A few studies have addressed a model of this nature assuming some degree of dependence between the two random variables (and most of these studies include copulas). In this paper, we focus on the collective and Bayes net premiums for the aggregate amount of claims under a compound model assuming some degree of dependence between the random variables Θ₁ and Θ₂. The degree of dependence is modelled using the Sarmanov–Lee family of distributions [Sarmanov, O.V., 1966. Generalized normal correlation and two-dimensional Frechet classes. Doklady (Soviet Mathematics) 168, 596–599 and Ting-Lee, M.L., 1996. Properties and applications of the Sarmanov family of bivariate distributions. Communications Statistics: Theory and Methods 25 (6) 1207–1222], which allows us to study the impact of this assumption on the collective and Bayes net premiums. The results obtained show that a low degree of correlation produces Bayes premiums that are highly sensitive.     

Diagnosability of regular systems

A novel approach aimed at evaluating the diagnosability of regular systems under the PMC model is introduced. The diagnosability is defined as the ability to provide a correct diagnosis, although possibly incomplete. This concept is somehow intermediate between one-step diagnosability and sequential diagnosability. A lower bound to diagnosability is determined by lower bounding the minimum of a “syndrome-dependent” bound t_σ over the set of all the admissible syndromes. In turn, t_σ is determined by evaluating the cardinality of the smallest consistent fault set containing an aggregate of maximum cardinality. The new approach, which applies to any regular system, relies on the “edge-isoperimetric inequalities” of connected components of units declaring each other non-faulty. This approach has been used to derive tight lower bounds to the diagnosability of toroidal grids and hypercubes, which improve the existing bounds for the same structures.     

A geometric proof of the combinatorial bounds for the number of optimal solutions for the Euclidean 2-center problem

We provide lower and upper bounds for γ(n), the number of optimal solutions for the two-center problem: “Given a set S of n points in the real plane, find two closed discs whose union contains all of the points such that the radius of the larger disc is minimized.”The main result of the paper shows the matching upper and lower bounds for the two-center problem and demonstrates that γ(n)=n.     

Self-organizing lists and independent references: A statistical synergy

Let R₁,…, R_n be a linear list of n elements. We assume the independent reference model, with a fixed but unknown access probability vector. We survey briefly the problem of reorganizing the list dynamically, on the basis of accrued references, with the objective of minimizing the expected access cost. The counter scheme (CS) is known to be asymptotically optimal for this purpose. The paper explores the CS, with the aim of reducing its storage requirements. We start with a detailed exposition of its cost function and then point out that it interacts with the access model to produce some remarkable synergistic effects. These make it possible to use very effective “truncated versions” of the CS, which have very modest space requirements. The versions we consider are: (i) the “limited-counters scheme”, which bounds each of the frequency counters to a maximal value c; (ii) the original CS with a bound on the number of references during which the scheme is active. The bound is chosen so as to achieve a desired level of performance compared with the optimal policy.     

The Enumeration of Permutations with a Prescribed Number of “Forbidden” Patterns

We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of “forbidden” patterns that seems to indicate that the enumerating sequence is always P-recursive. We illustrate the method completely in terms of the patterns “abc,” “cab,” and “abcd.”     

Random creation and dispersion of mass

Consider evolution of density of a mass or a population, geographically situated in a compact region of space, assuming random creation-annihilation and migration, or dispersion of mass, so the evolution is a random measure. When the creation-annihilation and dispersion are diffusions the situation is described formally by a stochastic partial differential equation; ignoring dispersion make approximations to the initial density by atomic measures and if the corresponding discrete random measures converge “in law” to a unique random measure call it a solution. To account for dispersion Trotter's product formula is applied to semiflows corresponding to dispersion and creation-annihilation. Existence of solutions has been a conjecture for several years despite a claim in ([2], J. Multivariate Anal. 5, 1–52). We show that solutions exist and that non-deterministic solutions are “smeared” continuous-state branching diffusions.     

The determination of quasistationary distributions directly from the transition rates of an absorbing Markov chain

There are many stochastic systems arising in areas as diverse as wildlife management, chemical kinetics and reliability theory, which eventually “die out,” yet appear to be stationary over any reasonable time scale. The notion of a quasistationary distribution has proved to be a potent tool in modelling this behaviour. In finite-state systems the existence of a quasistationary distribution is guaranteed. However, in the infinite-state case this may not always be so and the question of whether or not quasistationary distributions exist requires delicate mathematical analysis. The purpose of this paper is to present simple conditions for the existence of quasistationary distributions for continuous-time Markov chains and to demonstrate how these can be applied in practice.     

CDS simulation and pattern formation of phase separation

Several properties of the generation and evolution of phase separating patterns for binary material studied by CDS model are proposed. The main conclusions are (1) for alloys spinodal decomposition, the conceptions of “macro-pattern” and “micropattern” are posed by “black-and- white graph” and “gray-scale graph” respectively. We find that though the four forms of map f that represent the self-evolution of order parameter in a cell (lattice) are similar to each other in “macro-pattern”, there are evident differences in their micro-pattern, e.g., some different fine netted structures in the black domain and the white domain are found by the micro-pattern, so that distinct mechanical and physical behaviors shall be obtained. (2) If the two constitutions of block copolymers are not symmetric (i.e. r ≠ 0.5), a pattern called “grain-strip cross pattern” is discovered, in the 0.43 <r <0.45.     

Multiway Dependence in Exponential Family Conditional Distributions

Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the “pairwise-only dependence” assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192–225) formulated exponential family “auto-models” and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies.