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

Convergence of Probability Measures

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

A rich and flexible class of random probability measures, which we call stick-breaking priors, can be constructed using a sequence of independent beta random variables. Examples of random measures that have this characterization include the Dirichlet process, its two-parameter extension, the two-parameter Poisson–Dirichlet process, finite dimensional Dirichlet priors, and beta two-parameter processes. The rich nature of stick-breaking priors offers Bayesians a useful class of priors for nonparametric problems, while the similar construction used in each prior can be exploited to develop a general computational procedure for fitting them. In this article we present two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stick-breaking priors. The first type of Gibbs sampler, referred to as a Polya urn Gibbs sampler, is a generalized version of a widely used Gibbs sampling method currently employed for Dirichlet process computing. This method applies t...

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Gibbs sampling methods for stick-breaking priors

Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.

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Combinatorial Stochastic Processes

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

Stochastic Processes for Insurance and Finance

Concepts from Insurance and Finance. Probability Distributions. Premiums and Ordering of Risks. Distributions of Aggregate Claim Amount. Risk Processes. Renewal Processes and Random Walks. Markov Chains. Continuous-Time Markov Models. Martingale Techniques I. Martingale Techniques II. Piecewise Deterministic Markov Processes. Point Processes. Diffusion Models. Distribution Tables. References. Index.

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On random discrete distributions

We consider a Markov process $X(t)$ with extended generator ${\mathbf{A}}$ and domain $\cal{D}({\mathbf{A}})$. Let $\{{\cal F}_t\}$ be a right-continuous history filtration and ${\mathbb{P}}_t$ denote the restriction of ${\mathbb{P}}$ to ${\cal F}_t$. Let $\tilde{{\mathbb{P}}}$ be another probability measure on $(\Omega,{\cal F})$ such that $\rm d\tilde{{\mathbb{P}}}_t/\rm d{\mathbb{P}}_t=E^h(t)$, where $$ E^h(t)=\frac{h(X(t))}{h(X(0))}\exp \left(-\int_0^t \frac{({\mathbf{A}}h)(X(s))}{h(X(s))}\rm ds \right) $$ is a true martingale for a positive function $h\in\cal{D}({\mathbf{A}})$. We demonstrate that the process $X(t)$ is a Markov process on the probability space $ (\Omega, {\cal F}, \{{\cal F}_t\},\tilde{{\mathbb{P}}}) $, we find its extended generator $\tilde{{\mathbf{A}}}$ and provide sufficient conditions under which $\cal{D}(\tilde{{\mathbf{A}}})=\cal{D}({\mathbf{A}})$. We apply this result to continuous-time Markov chains, to piecewise deterministic Markov processes and to diffusion processes (in this case a special choice of $h$ yields the classical Cameron--Martin--Girsanov theorem).

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A technique for exponential change of measure for Markov processes

This paper is concerned with the numerical computation of the probability ψ( u ) of ruin with initial reserve u . The basic assumption states that the claim size distribution is phase-type in the sense of Neuts. The models considered are: the classical compound Poisson risk process, the Sparre Anderse process and varying environments which are either governed by a Markov process or exhibit periodic fluctuations. The computational steps involve the iterative solution of a non-linear matrix equation Q = Ψ ( Q ) as well as the evaluation of matrix-exponentials e Qu . A number of worked-out numerical examples are presented.

Tomasz Rolski

Papers

Stochastic Processes for Insurance and Finance

Stochastic Processes for Insurance and Finance

On random discrete distributions

A technique for exponential change of measure for Markov processes

Computational methods in risk theory: A matrix-algorithmic approach