Numerical Methods for Structured Markov Chains

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1 University Press Scholarship Online You are looking at 1-10 of 34 items for: keywords : stochastic processes Stochastic Methods in Neuroscience Carlo Laing and Gabriel J Lord (eds) Published in print: 2009 Published Online: February 2010 ISBN: eisbn: Item type: book acprof:oso/ We give a brief introduction to modelling in mathematical neuroscience, to stochastic processes, and stochastic differential equations as well as an overview of the book. Mathematical Preliminaries Partha P. Mitra and Hemant Bokil in Observed Brain Dynamics Published in print: 2007 Published Online: May 2009 ISBN: eisbn: acprof:oso/ This chapter reviews a broad range of mathematical topics relevant to the rest of the book. It begins with a brief discussion of real and complex numbers and elementary real and complex functions, followed by a summary of linear algebra, paying special attention to matrix decomposition techniques. Fourier analysis is discussed in some detail because this is a topic of central importance to time series analysis. After a brief review of probability theory, the core set of topics for the chapter is considered, dealing with stochastic process theory. This includes a discussion of point as well as continuous processes. Numerical Methods for Structured Markov Chains Dario A. Bini, Guy Latouche, and Beatrice Meini Published in print: 2005 Published Online: September 2007 ISBN: eisbn: Item type: book acprof:oso/ Page 1 of 5

2 The book deals with the numerical solution of structured Markov chains which include M/G/1 and G/M/1-type Markov chains, QBD processes, non-skip-free queues, and tree-like stochastic processes and has a wide applicability in queueing theory and stochastic modeling. It presents in a unified language the most up to date algorithms, which are so far scattered in diverse papers, written with different languages and notation. It contains a thorough treatment of numerical algorithms to solve these problems, from the simplest to the most advanced and most efficient. Nonlinear matrix equations are at the heart of the analysis of structured Markov chains, they are analysed both from the theoretical, from the probabilistic, and from the computational point of view. The set of methods for solution contains functional iterations, doubling methods, logarithmic reduction, cyclic reduction, and subspace iteration, all are described and analysed in detail. They are also adapted to interesting specific queueing models coming from applications. The book also offers a comprehensive and self-contained treatment of the structured matrix tools which are at the basis of the fastest algorithmic techniques for structured Markov chains. Results about Toeplitz matrices, displacement operators, and Wiener-Hopf factorizations are reported to the extent that they are useful for the numerical treatment of Markov chains. Every and all solution methods are reported in detailed algorithmic form so that they can be coded in a high-level language with minimum effort. Stochastic Processes and Stochastic Calculus Claus Munk in Fixed Income Modelling ISBN: eisbn: acprof:oso/ The price of an asset at a future point in time will typically be unknown, i.e. a random variable. In order to describe the uncertain evolution in the price of the asset over time, we need a collection of random variables, namely one random variable for each point in time. Such a collection of random variables is called a stochastic process. Modern finance models therefore apply stochastic processes to represent the evolution in prices as well as interest rates and other relevant quantities over time. This is also the case for the dynamic interest rate models presented in this book. This chapter gives an introduction to stochastic processes and the mathematical tools needed to do calculations with stochastic processes, the so-called stochastic calculus, focusing on processes and results that will become important in later chapters. Page 2 of 5

3 Stochastic Integrals Tomas Björk in Arbitrage Theory in Continuous Time Published in print: 2004 Published Online: October 2005 ISBN: eisbn: Page 3 of This chapter discusses the modelling of asset prices as continuous time stochastic processes. Diffusion processes and stochastic differential equations are used as building blocks to obtain the most complete and elegant theory. Practice exercises are included. A Brief Introduction to Some Simple Stochastic Processes Benjamin Lindner in Stochastic Methods in Neuroscience Published in print: 2009 Published Online: February 2010 ISBN: eisbn: acprof:oso/ This chapter gives an overview of simple continuous, two-state, and point processes playing a role in theoretical neuroscience. First, various characteristics of these stochastic processes are introduced such as probability densities, moments, correlation functions, the correlation time, and the noise intensity of a process. Then analytical and numerical methods to calculate or compute these various statistics are explained and illustrated by means of simple examples (Ornstein Uhlenbeck process, random telegraph noise, Poissonian shot noise). Further, useful relations among the different statistics (Wiener Khinchin theorem, relations between spectral and interval statistics of point processes) are also discussed. Classical Probability Theory and Stochastic Processes Heinz-Peter Breuer and Francesco Petruccione in The Theory of Open Quantum Systems Published in print: 2007 Published Online: January 2010 ISBN: eisbn: acprof:oso/ This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental

4 concepts of probability space and Kolmogorov axioms. These concepts are then used to define random variables and stochastic processes. The mathematical formulation of the special class of Markov processes through classical master equations is given, including deterministic processes (Liouville equation), jump processes (Pauli master equation), and diffusion processes (Fokker Planck equation). Special stochastic processes which play an important role in the developments of the following chapters, such as piecewise deterministic processes and Lévy processes, are described in detail together with their basic physical properties and various mathematical formulations in terms of master equations, path integral representation, and stochastic differential equations. Introduction Eric Renshaw in Stochastic Population Processes: Analysis, Approximations, Simulations ISBN: eisbn: acprof:oso/ This introductory chapter presents a summary of the topics covered which not only provides a road map to specific content but also highlights the strong relationships which exist between the various processes under study. It covers simple stochastic processes, single-species population dynamics, bivariate populations, and spatial-temporal processes. Representing Response Times as Random Variables R. Duncan Luce in Response Times: Their Role in Inferring Elementary Mental Organization Published in print: 1991 Published Online: January 2008 ISBN: eisbn: acprof:oso/ This chapter begins with a discussion of the study of response times. Response times are treated as observations of a random variable. The mathematics of stochastic processes is then used to understand the process. This gives rise to the distributions of these random variables. Generating functions and elementary concepts of stochastic processes are discussed. Page 4 of 5

5 Spatial processes Eric Renshaw in Stochastic Population Processes: Analysis, Approximations, Simulations ISBN: eisbn: acprof:oso/ All the previous analyses are based on the assumption that populations develop at a single site where individuals mix homogeneously. Whilst this is mathematically ideal, in that it facilitates theoretical development, in reality there are many situations in which it may be violated. For not only may a population be spatially distributed across several interlinked sites, but even within a specific site the chance of two individuals meeting and interacting may well depend on the distance between them. Although this fact was realized early on in the development of theoretical population dynamics, the high degree of mathematical intractability which rides along with it has meant that little analytic progress has been made relative to non-spatial scenarios. This chapter exposes the underlying theoretical difficulties, highlights directions in which some degree of progress can be made, and shows that the introduction of space generates a whole new concept of a stochastic dynamic. In this latter construct, single-site processes, which on their own result in early extinction, can generate long-term persistence when linked together. Page 5 of 5