This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Diffusions, Markov processes, and martingales

Levy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance is justified by their application in many areas of classical and modern stochastic models.

This textbook forms the basis of a graduate course on the theory and applications of Levy processes, from the perspective of their path fluctuations. Central to the presentation are decompositions of the paths of Levy processes in terms of their local maxima and an understanding of their short- and long-term behaviour.

The book aims to be mathematically rigorous while still providing an intuitive feel for underlying principles. The results and applications often focus on the case of Levy processes with jumps in only one direction, for which recent theoretical advances have yielded a higher degree of mathematical transparency and explicitness. Each chapter has a comprehensive set of exercises with complete solutions.

Introductory Lectures on Fluctuations of Lévy Processes with Applications

The Brunn-Minkowski inequality in Gauss space

Measure-valued Markov processes

The most common model to support workforce management of telephone call centers is theM/ M/ N/ B model, in particular its special casesM/ M/ N (Erlang C, which models out busy signals) andM/ M/ N/ N (Erlang B, disallowing waiting). All of these models lack a central prevalent feature, namely, that impatient customers might decide to leave (abandon) before their service begins.In this paper, we analyze the simplest abandonment model, in which customers' patience is exponentially distributed and the system's waiting capacity is unlimited ( M/ M/ N +M). Such a model is both rich and analyzable enough to provide information that is practically important for call-center managers. We first outline a method for exact analysis of theM/ M/ N +M model, that while numerically tractable is not very insightful. We then proceed with an asymptotic analysis of theM/ M/ N +M model, in a regime that is appropriate for large call centers (many agents, high efficiency, high service level). Guided by the asymptotic behavior, we derive approximations for performance measures and propose "rules of thumb" for the design of large call centers. We thus add support to the growing acknowledgment that insights from diffusion approximations are directly applicable to management practice.

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Designing a Call Center with Impatient Customers

Markov Processes and Potential Theory

Preliminaries.- Resolvents.- Ray resolvents and semigroups.- Increasing sequences of supermartingales.- Processes.- Processes continued.- Characterization of previsible stopping times.- Some topology and measure theory.- Right processes.- The ray knight compactification.- Comparison of processes.- Right processes continued: Shih's theorem.- Comparison of (Xt?) and (X t? * ).- U-spaces.- The ray space.

Ronald K. Getoor

Papers

Markov Processes and Potential Theory

Markov Processes: Ray Processes and Right Processes

Naturality, standardness, and weak duality for Markov processes

The asymptotic distribution of the eigenvalues for a class of Markov operators.

Excursions of Brownian motion and bessel processes