Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

This paper analyzes a family of multivariate point process models of correlated event timing whose arrival intensity is driven by an affine jump diffusion. The components of an affine point process are self- and cross-exciting and facilitate the description of complex event dependence structures. ODEs characterize the transform of an affine point process and the probability distribution of an integer-valued affine point process. The moments of an affine point process take a closed form. This guarantees a high degree of computational tractability in applications. We illustrate this in the context of portfolio credit risk, where the correlation of corporate defaults is the main issue. We consider the valuation of securities exposed to correlated default risk and demonstrate the significance of our results through market calibration experiments. We show that a simple model variant can capture the default clustering implied by index and tranche market prices during September 2008, a month that witnessed significant volatility.

https://epubs.siam.org/doi/pdf/10.1137/090771272

Affine Point Processes and Portfolio Credit Risk

This paper analyzes a family of multivariate point process models of correlated event timing whose arrival intensity is driven by an affine jump diffusion. The components of an affine point process are self- and cross-exciting, and facilitate the description of complex event dependence structures. Ordinary differential equations characterize the transform of an affine point process and the probability distribution of an integer-valued affine point process. The moments of an affine point process take a closed form. This guarantees a high degree of computational tractability in applications. We illustrate this in the context of portfolio credit risk, where the correlation of corporate defaults is the main issue. We consider the valuation of securities exposed to correlated default risk, and demonstrate the significance of our results through market calibration experiments. We show that a simple model variant can capture the default clustering implied by index and tranche market prices during September 2008, a month that witnessed significant volatility.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1625099_code295933.pdf?abstractid=908045&mirid=1&type=2

/pdf/a-bibliography-of-publications-about-the-fortran-programming-q3kg2hu189.pdf

A Bibliography of Publications about the Fortran Programming Language: Part 1: 1956{1980

Modeling and performance prediction are becoming increasingly important issues in the design and operation of computer communications systems. Complexities in their configuration and sophistications in resource sharing found in today's computer communications demand our intensive effort to enhance the modeling capability. The present paper is intended to review the state of affairs of analytic methods, queueing analysis techniques in particular, which are essential to modeling of computer communication systems. First we review basic properties of exponential queueing systems, and then give an overview of recent progress made in the areas of queueing network models and discrete-time queueing systems. A unified treatment of buffer storage overflow problems will be discussed as an application example, in which we call attention to the analogy between buffer behavior and waiting time in the GI/G/1 queue. Another application deals with the analysis of various multiplexing techniques and network configuration. An extensive reference list of the subject fields is also provided.

/pdf/queueing-models-for-computer-communications-system-analysis-4k09i67nnq.pdf

Queueing Models for Computer Communications System Analysis

This paper deals with the stationary analysis of the finite, single server queue in discrete time. The following stntionary distributions and other quantities of practical interest are investigated: (1) the joint density of the queue length and the residual service time, (2) the queue length distribution and its mean, (3) the distribution of the residual service time and its mean, (4) the distribution and the expected value of the number of customers lost per unit of time due to saturation of the waiting capacity, (5) the distribution and the mean of the waiting time, (6) the asymptotic distribution of the queue length following departures



The latter distribution is particularly noteworthy, in view of the substantial difference which exists, in general, between the distributions of the queue lengths at arbitrary points of time and those immediately following departures.

The single server queue in discrete time-numerical analysis II

The paper extends the ergodic theorems of information theory (Shannon-MacMillan-Breiman theorems) to spaces with an infinite invariant measure. An L1 difference theorem and a pointwisa ratio theorem are proved, for the information of spreading partitions. For the validity of the theorems it is assumed that the supremum f* of the conditional information given the increasing “past” is integrable. Simple necessary and sufficient conditions for the integrability of f* are obtained in special cases: If the initial partition is composed of one state of a null-recurrent Markov chain, then f* is integrable if and only if the partition of this state according to the first return times has finite entropy.

On convergence of information in spaces with infinite invariant measure

This paper presents a fast algorithm for generating the-non zero indices used in Faa di Bruno's formula for the higher order derivatives of compositions of functions. The application of this formula to the solution of functional equations and to some problems in queueing theory is indicated.

An algorithm for calculating indices in Fàa di Bruno's formula

/pdf/an-operator-ergodic-theorem-for-sequences-of-functions-3iw3m9bcsj.pdf

An operator ergodic theorem for sequences of functions

In this note, a relationship is established between the entropy, defined by Krengel for an endomorphism of a σ-finite measure space, and the notion of a spreading partition. This relationship is used to answer in the “quasi-finite” case a question raised by Krengel concerning the entropy of the product endomorphism on the direct product of a finite and σ-finite measure space.

Eugene M. Klimko

Papers

The single server queue in discrete time-numerical analysis II

On convergence of information in spaces with infinite invariant measure

An algorithm for calculating indices in Fàa di Bruno's formula

An operator ergodic theorem for sequences of functions

On the entropy of a product endomorphism in infinite measure spaces