Carlo Laing and Gabriel J Lord (eds)
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199235070
- eISBN:
- 9780191715778
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199235070.001.0001
- Subject:
- Mathematics, Biostatistics
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.
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.
Dario A. Bini, Guy Latouche, and Beatrice Meini
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198527688
- eISBN:
- 9780191713286
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527688.001.0001
- Subject:
- Mathematics, Numerical Analysis
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 ...
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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.Less
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.
Partha P. Mitra and Hemant Bokil
- Published in print:
- 2007
- Published Online:
- May 2009
- ISBN:
- 9780195178081
- eISBN:
- 9780199864829
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195178081.003.0005
- Subject:
- Neuroscience, Techniques, Molecular and Cellular Systems
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, ...
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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.Less
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.
Richard M. Murray
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161532
- eISBN:
- 9781400850501
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161532.003.0004
- Subject:
- Biology, Biochemistry / Molecular Biology
This chapter explores stochastic behavior in biomolecular systems. It does so by first building on the preliminary discussion of stochastic modeling laid out in Chapter 2. The chapter reviews methods ...
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This chapter explores stochastic behavior in biomolecular systems. It does so by first building on the preliminary discussion of stochastic modeling laid out in Chapter 2. The chapter reviews methods for modeling stochastic processes, including the chemical master equation (CME), the chemical Langevin equation (CLE), and the Fokker–Planck equation (FPE). Given a stochastic description, the chapter then analyzes the behavior of the system using a collection of stochastic simulation and analysis tools. This chapter makes use of a variety of topics in stochastic processes; readers should have a good working knowledge of basic probability and some exposure to simple stochastic processes.Less
This chapter explores stochastic behavior in biomolecular systems. It does so by first building on the preliminary discussion of stochastic modeling laid out in Chapter 2. The chapter reviews methods for modeling stochastic processes, including the chemical master equation (CME), the chemical Langevin equation (CLE), and the Fokker–Planck equation (FPE). Given a stochastic description, the chapter then analyzes the behavior of the system using a collection of stochastic simulation and analysis tools. This chapter makes use of a variety of topics in stochastic processes; readers should have a good working knowledge of basic probability and some exposure to simple stochastic processes.
Benjamin Lindner
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199235070
- eISBN:
- 9780191715778
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199235070.003.0001
- Subject:
- Mathematics, Biostatistics
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 ...
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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.Less
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.
Claus Munk
- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199575084
- eISBN:
- 9780191728648
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199575084.003.0003
- Subject:
- Economics and Finance, Financial Economics
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 ...
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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.Less
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.
Tomas Björk
- Published in print:
- 2004
- Published Online:
- October 2005
- ISBN:
- 9780199271269
- eISBN:
- 9780191602849
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199271267.003.0004
- Subject:
- Economics and Finance, Financial Economics
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 ...
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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.Less
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.
Heinz-Peter Breuer and Francesco Petruccione
- Published in print:
- 2007
- Published Online:
- February 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.003.01
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental concepts of probability space and Kolmogorov axioms. These ...
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This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental 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.Less
This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental 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.
Andrew J. Connolly, Jacob T. VanderPlas, Alexander Gray, Andrew J. Connolly, Jacob T. VanderPlas, and Alexander Gray
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691151687
- eISBN:
- 9781400848911
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691151687.003.0010
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter summarizes the fundamental concepts and tools for analyzing time series data. Time series analysis is a branch of applied mathematics developed mostly in the fields of signal processing ...
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This chapter summarizes the fundamental concepts and tools for analyzing time series data. Time series analysis is a branch of applied mathematics developed mostly in the fields of signal processing and statistics. Contributions to this field, from an astronomical perspective, have predominantly focused on unevenly sampled data, low signal-to-noise data, and heteroscedastic errors. The chapter starts with a brief introduction to the main concepts in time series analysis. It then discusses the main tools from the modeling toolkit for time series analysis. Despite being set in the context of time series, many tools and results are readily applicable in other domains, and for this reason the examples presented will not be strictly limited to time-domain data. Armed with the modeling toolkit, the chapter goes on to discuss the analysis of periodic time series, search for temporally localized signals, and concludes with a brief discussion of stochastic processes.Less
This chapter summarizes the fundamental concepts and tools for analyzing time series data. Time series analysis is a branch of applied mathematics developed mostly in the fields of signal processing and statistics. Contributions to this field, from an astronomical perspective, have predominantly focused on unevenly sampled data, low signal-to-noise data, and heteroscedastic errors. The chapter starts with a brief introduction to the main concepts in time series analysis. It then discusses the main tools from the modeling toolkit for time series analysis. Despite being set in the context of time series, many tools and results are readily applicable in other domains, and for this reason the examples presented will not be strictly limited to time-domain data. Armed with the modeling toolkit, the chapter goes on to discuss the analysis of periodic time series, search for temporally localized signals, and concludes with a brief discussion of stochastic processes.
Michael Laver and Ernest Sergenti
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691139036
- eISBN:
- 9781400840328
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691139036.003.0004
- Subject:
- Political Science, Comparative Politics
This chapter develops the methods for designing, executing, and analyzing large suites of computer simulations that generate stable and replicable results. It starts with a discussion of the ...
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This chapter develops the methods for designing, executing, and analyzing large suites of computer simulations that generate stable and replicable results. It starts with a discussion of the different methods of experimental design, such as grid sweeping and Monte Carlo parameterization. Next, it demonstrates how to calculate mean estimates of output variables of interest. It does so by first discussing stochastic processes, Markov Chain representations, and model burn-in. It focuses on three stochastic process representations: nonergodic deterministic processes that converge on a single state; nondeterministic stochastic processes for which a time average provides a representative estimate of the output variables; and nondeterministic stochastic processes for which a time average does not provide a representative estimate of the output variables. The estimation strategy employed depends on which stochastic process the simulation follows. Lastly, the chapter presents a set of diagnostic checks used to establish an appropriate sample size for the estimation of the means.Less
This chapter develops the methods for designing, executing, and analyzing large suites of computer simulations that generate stable and replicable results. It starts with a discussion of the different methods of experimental design, such as grid sweeping and Monte Carlo parameterization. Next, it demonstrates how to calculate mean estimates of output variables of interest. It does so by first discussing stochastic processes, Markov Chain representations, and model burn-in. It focuses on three stochastic process representations: nonergodic deterministic processes that converge on a single state; nondeterministic stochastic processes for which a time average provides a representative estimate of the output variables; and nondeterministic stochastic processes for which a time average does not provide a representative estimate of the output variables. The estimation strategy employed depends on which stochastic process the simulation follows. Lastly, the chapter presents a set of diagnostic checks used to establish an appropriate sample size for the estimation of the means.
R. Duncan Luce
- Published in print:
- 1991
- Published Online:
- January 2008
- ISBN:
- 9780195070019
- eISBN:
- 9780199869879
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195070019.003.0001
- Subject:
- Psychology, Cognitive Models and Architectures
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 ...
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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.Less
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.
Eric Renshaw
- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199575312
- eISBN:
- 9780191728778
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199575312.003.0001
- Subject:
- Mathematics, Applied Mathematics, Mathematical Biology
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 ...
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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.Less
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.
David F. Hendry
- Published in print:
- 1995
- Published Online:
- November 2003
- ISBN:
- 9780198283164
- eISBN:
- 9780191596384
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198283164.003.0002
- Subject:
- Economics and Finance, Econometrics
The main concepts for empirical modelling of economic time series are explained: parameter and parameter space; constancy; structure; distributional shape; identification and observational ...
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The main concepts for empirical modelling of economic time series are explained: parameter and parameter space; constancy; structure; distributional shape; identification and observational equivalence; interdependence; stochastic process; conditioning; white noise; autocorrelation; stationarity; integratedness; trend; heteroscedasticity; dimensionality; aggregation; sequential factorization; and marginalization. A formal data‐generation process (DGP) for economics is the joint data density with an innovation error. Empirical models derive from reduction operations applied to the DGP.Less
The main concepts for empirical modelling of economic time series are explained: parameter and parameter space; constancy; structure; distributional shape; identification and observational equivalence; interdependence; stochastic process; conditioning; white noise; autocorrelation; stationarity; integratedness; trend; heteroscedasticity; dimensionality; aggregation; sequential factorization; and marginalization. A formal data‐generation process (DGP) for economics is the joint data density with an innovation error. Empirical models derive from reduction operations applied to the DGP.
Robert M. Mazo
- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199556441
- eISBN:
- 9780191705625
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556441.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
Brownian motion is the incessant motion of small particles immersed in an ambient medium. It is due to fluctuations in the motion of the medium particles on the molecular scale. The name has been ...
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Brownian motion is the incessant motion of small particles immersed in an ambient medium. It is due to fluctuations in the motion of the medium particles on the molecular scale. The name has been carried over to other fluctuation phenomena. This book treats the physical theory of Brownian motion. The extensive mathematical theory, which treats the subject as a subfield of the general theory of random processes, is touched on but not presented in any detail. Random or stochastic process theory and statistical mechanics are the primary tools. The first eight chapters treat the stochastic theory and some applications. The next six present the statistical mechanical point of view. Then follows chapters on applications to diffusion, noise, and polymers, followed by a treatment of the motion of interacting Brownian particles. The book ends with a final chapter treating simulation, fractals, and chaos.Less
Brownian motion is the incessant motion of small particles immersed in an ambient medium. It is due to fluctuations in the motion of the medium particles on the molecular scale. The name has been carried over to other fluctuation phenomena. This book treats the physical theory of Brownian motion. The extensive mathematical theory, which treats the subject as a subfield of the general theory of random processes, is touched on but not presented in any detail. Random or stochastic process theory and statistical mechanics are the primary tools. The first eight chapters treat the stochastic theory and some applications. The next six present the statistical mechanical point of view. Then follows chapters on applications to diffusion, noise, and polymers, followed by a treatment of the motion of interacting Brownian particles. The book ends with a final chapter treating simulation, fractals, and chaos.
Eric Renshaw
- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199575312
- eISBN:
- 9780191728778
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199575312.003.0009
- Subject:
- Mathematics, Applied Mathematics, Mathematical Biology
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 ...
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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.Less
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.
Melvin Lax, Wei Cai, and Min Xu
- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
A random or stochastic process is a random variable that evolves in time by some random mechanism (of course, the time variable can be replaced by a space variable, or some other variable, in ...
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A random or stochastic process is a random variable that evolves in time by some random mechanism (of course, the time variable can be replaced by a space variable, or some other variable, in application). The variable can have a discrete set of values at a given time, or a continuum of values may be available. Likewise, the time variable can be discrete or continuous. A stochastic process is regarded as completely described if the probability distribution is known for all possible sets of times. A stationary process is one which has no absolute time origin. All probabilities are independent of a shift in the origin of time. This chapter discusses multitime probability description, conditional probabilities, stationary, Gaussian, and Markovian processes, and the Chapman–Kolmogorov condition.Less
A random or stochastic process is a random variable that evolves in time by some random mechanism (of course, the time variable can be replaced by a space variable, or some other variable, in application). The variable can have a discrete set of values at a given time, or a continuum of values may be available. Likewise, the time variable can be discrete or continuous. A stochastic process is regarded as completely described if the probability distribution is known for all possible sets of times. A stationary process is one which has no absolute time origin. All probabilities are independent of a shift in the origin of time. This chapter discusses multitime probability description, conditional probabilities, stationary, Gaussian, and Markovian processes, and the Chapman–Kolmogorov condition.
Claus Munk
- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199585496
- eISBN:
- 9780191751790
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199585496.003.0002
- Subject:
- Economics and Finance, Econometrics
Uncertainty is a key component of financial markets and thus of any asset pricing theory. This chapter provides the tools from probability theory that are being used in the asset pricing models ...
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Uncertainty is a key component of financial markets and thus of any asset pricing theory. This chapter provides the tools from probability theory that are being used in the asset pricing models covered in the remaining part of the book. The mathematical representation of uncertainty and information flow is explained. Stochastic processes are introduced with numerous examples both in discrete time and in continuous time. The important Ito’s Lemma is presented and illustrated by examples. The simultaneous handling of multiple stochastic processes is also discussed. The chapter is accessible with only little prior exposure to probability theory and continuous-time finance models.Less
Uncertainty is a key component of financial markets and thus of any asset pricing theory. This chapter provides the tools from probability theory that are being used in the asset pricing models covered in the remaining part of the book. The mathematical representation of uncertainty and information flow is explained. Stochastic processes are introduced with numerous examples both in discrete time and in continuous time. The important Ito’s Lemma is presented and illustrated by examples. The simultaneous handling of multiple stochastic processes is also discussed. The chapter is accessible with only little prior exposure to probability theory and continuous-time finance models.
Anindya Banerjee, Juan J. Dolado, John W. Galbraith, and David F. Hendry
- Published in print:
- 1993
- Published Online:
- November 2003
- ISBN:
- 9780198288107
- eISBN:
- 9780191595899
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198288107.003.0001
- Subject:
- Economics and Finance, Econometrics
Serves as an introductory overview for the rest of the book, and outlines its main aims. As a basis for the following chapters, an overview and clarification of equilibrium relationships in economic ...
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Serves as an introductory overview for the rest of the book, and outlines its main aims. As a basis for the following chapters, an overview and clarification of equilibrium relationships in economic theory is presented. A preliminary discussion of testing for orders of integration and the estimation of long‐run relationships is provided. The chapter summarizes key concepts from time‐series analysis and the theory of stochastic processes and, in particular, the theory of Brownian motion processes. Several empirical examples are offered as illustration of these concepts.Less
Serves as an introductory overview for the rest of the book, and outlines its main aims. As a basis for the following chapters, an overview and clarification of equilibrium relationships in economic theory is presented. A preliminary discussion of testing for orders of integration and the estimation of long‐run relationships is provided. The chapter summarizes key concepts from time‐series analysis and the theory of stochastic processes and, in particular, the theory of Brownian motion processes. Several empirical examples are offered as illustration of these concepts.
James Davidson
- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.001.0001
- Subject:
- Economics and Finance, Econometrics
This book aims to introduce modern asymptotic theory to students and practitioners of econometrics. It falls broadly into two parts. The first half provides a handbook and reference for the ...
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This book aims to introduce modern asymptotic theory to students and practitioners of econometrics. It falls broadly into two parts. The first half provides a handbook and reference for the underlying mathematics (Part I, Chapters 1‐6), statistical theory (Part II, Chapters 7‐11) and stochastic process theory (Part III, Chapters 12‐17). The second half provides a treatment of the main convergence theorems used in analysing the large sample behaviour of econometric estimators and tests. These are the law of large numbers (Part IV, Chapters 18‐21), the central limit theorem (Part V, Chapters 22‐25) and the functional central limit theorem (Part VI, Chapters 26‐30). The focus in this treatment is on the nonparametric approach to time series properties, covering topics such as nonstationarity, mixing, martingales, and near‐epoch dependence. While the approach is not elementary, care is taken to keep the treatment self‐contained. Proofs are provided for almost all the results.Less
This book aims to introduce modern asymptotic theory to students and practitioners of econometrics. It falls broadly into two parts. The first half provides a handbook and reference for the underlying mathematics (Part I, Chapters 1‐6), statistical theory (Part II, Chapters 7‐11) and stochastic process theory (Part III, Chapters 12‐17). The second half provides a treatment of the main convergence theorems used in analysing the large sample behaviour of econometric estimators and tests. These are the law of large numbers (Part IV, Chapters 18‐21), the central limit theorem (Part V, Chapters 22‐25) and the functional central limit theorem (Part VI, Chapters 26‐30). The focus in this treatment is on the nonparametric approach to time series properties, covering topics such as nonstationarity, mixing, martingales, and near‐epoch dependence. While the approach is not elementary, care is taken to keep the treatment self‐contained. Proofs are provided for almost all the results.
Robert M. Mazo
- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199556441
- eISBN:
- 9780191705625
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556441.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter extends the ideas of Chapter 2 by introducing the notion of stochastic process or random process and its distribution functions. The idea of a Markov process is defined. The ...
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This chapter extends the ideas of Chapter 2 by introducing the notion of stochastic process or random process and its distribution functions. The idea of a Markov process is defined. The Chapman–Kolmagorov equation for Markov processes is given. From this, the Fokker–Planck equation for homogeneous continuous Markov processes is derived. The calculus of stochastic processes is then discussed, i.e., questions of what is meant by convergence, continuity, integration, Fourier analysis. The chapter concludes with a short discussion of white noise, a completely random process.Less
This chapter extends the ideas of Chapter 2 by introducing the notion of stochastic process or random process and its distribution functions. The idea of a Markov process is defined. The Chapman–Kolmagorov equation for Markov processes is given. From this, the Fokker–Planck equation for homogeneous continuous Markov processes is derived. The calculus of stochastic processes is then discussed, i.e., questions of what is meant by convergence, continuity, integration, Fourier analysis. The chapter concludes with a short discussion of white noise, a completely random process.
