## Markov Processes

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0004
- Subject:
- Mathematics, Probability / Statistics

This chapter deals with Markov processes. It first defines the “Markov property” and shows that all the relevant information about a Markov process assuming values in a finite set of cardinality n ... More

## Hidden Markov Processes: The Complete Realization Problem

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0007
- Subject:
- Mathematics, Probability / Statistics

This chapter considers hidden Markov processes (HMPs), focusing on the so-called complete realization problem. It is quite easy to prove a universal necessary condition for the given process to have ... More

## Hidden Markov Processes: Basic Properties

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0006
- Subject:
- Mathematics, Probability / Statistics

This chapter considers the basic properties of hidden Markov processes (HMPs) or hidden Markov models (HMMs), a special type of stochastic process. It begins with a discussion of three distinct types ... More

## Profit scoring and dynamic models

*Lyn C. Thomas*

### in Consumer Credit Models: Pricing, Profit and Portfolios

- Published in print:
- 2009
- Published Online:
- May 2009
- ISBN:
- 9780199232130
- eISBN:
- 9780191715914
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232130.003.0004
- Subject:
- Mathematics, Applied Mathematics, Mathematical Finance

This chapter begins by reviewing the role of behavioural scoring and risk/reward matrices in the way a lender manages borrowers. It points out that current methods do not allow for the future changes ... More

## Hidden Markov Processes: Theory and Applications to Biology

*M. Vidyasagar*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.001.0001
- Subject:
- Mathematics, Probability / Statistics

This book explores important aspects of Markov and hidden Markov processes and the applications of these ideas to various problems in computational biology. It starts from first principles, so that ... More

## The Theory of Open Quantum Systems

*Heinz-Peter Breuer and Francesco Petruccione*

- Published in print:
- 2007
- Published Online:
- February 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This book treats the central physical concepts and mathematical techniques used to investigate the dynamics of open quantum systems. To provide a self-contained presentation, the text begins with a ... More

## Some Applications to Computational Biology

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0008
- Subject:
- Mathematics, Probability / Statistics

This chapter considers some applications of Markov processes and hidden Markov processes to computational biology. It introduces three important problems, namely: sequence alignment, the gene-finding ... More

## 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:
- 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 ... More

## On Some Monotone Markov Models

*Mukul Majumdar and Nigar Hashimzade*

### in Dimensions of Economic Theory and Policy: Essays for Anjan Mukherji

- Published in print:
- 2011
- Published Online:
- September 2012
- ISBN:
- 9780198073970
- eISBN:
- 9780199081615
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198073970.003.0003
- Subject:
- Economics and Finance, Microeconomics

This chapter presents some results and examples on the invariant distributions of monotone Markov processes. It summarizes the fundamental results on the existence, uniqueness, and stability of the ... More

## STOCHASTIC PROCESSES

*Robert M. Mazo*

### in Brownian Motion: Fluctuations, Dynamics, and Applications

- 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 ... More

## Jump Markov Processes

*Gopinath Kallianpur and P. Sundar*

### in Stochastic Analysis and Diffusion Processes

- Published in print:
- 2014
- Published Online:
- April 2014
- ISBN:
- 9780199657063
- eISBN:
- 9780191781759
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199657063.003.0010
- Subject:
- Mathematics, Probability / Statistics, Applied Mathematics

After introducing Poisson processes, the form of their generators is explained. Jump Markov processes, generators, and explosion times are discussed. Skorohod space is introduced, and the proof of ... More

## Nonnegative Matrices

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0003
- Subject:
- Mathematics, Probability / Statistics

This chapter deals with nonnegative matrices, which are relevant in the study of Markov processes because the state transition matrix of such a process is a special kind of nonnegative matrix, known ... More

## Markov processes in continuous time and space

*Eric Renshaw*

### in Stochastic Population Processes: Analysis, Approximations, Simulations

- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199575312
- eISBN:
- 9780191728778
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199575312.003.0006
- Subject:
- Mathematics, Applied Mathematics, Mathematical Biology

Having dealt with processes in discrete space, and discrete and continuous time, this chapter investigates Markov processes in continuous space and time. It follows a natural progression through the ... More

## Quantum Master Equations

*Heinz-Peter Breuer and Francesco Petruccione*

### in The Theory of Open Quantum Systems

- Published in print:
- 2007
- Published Online:
- February 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.003.03
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces the fundamentals of the description of the quantum dynamics of open systems in terms of quantum master equations, together with its most important applications. Special ... More

## Properties of Langevin and Fokker–Planck equations

*Helmut Hofmann*

### in The Physics of Warm Nuclei: with Analogies to Mesoscopic Systems

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198504016
- eISBN:
- 9780191708480
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504016.003.0025
- Subject:
- Physics, Nuclear and Plasma Physics

This chapter introduces Langevin- and Fokker-Planck equations by way of a heuristic approach to the physics of Brownian particles. Variances of physical quantities are defined and related to ... More

## Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35)

*Zhen-Qing Chen and Masatoshi Fukushima*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691136059
- eISBN:
- 9781400840564
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691136059.001.0001
- Subject:
- Mathematics, Probability / Statistics

This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, the book ... More

## Diffusion in solids

*Rolf Hempelmann*

### in Quasielastic Neutron Scattering and Solid State Diffusion

- Published in print:
- 2000
- Published Online:
- January 2010
- ISBN:
- 9780198517436
- eISBN:
- 9780191706974
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198517436.003.0004
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter starts with a brief history of the science of diffusion. Macroscopically diffusion is characterized by various diffusion coefficients: the self-diffusion coefficient, the tracer ... More

## Stochastic Analysis and Diffusion Processes

*Gopinath Kallianpur and P Sundar*

- Published in print:
- 2014
- Published Online:
- April 2014
- ISBN:
- 9780199657063
- eISBN:
- 9780191781759
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199657063.001.0001
- Subject:
- Mathematics, Probability / Statistics, Applied Mathematics

Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. After proving the Doob-Meyer decomposition, quadratic variation processes and local ... More

## Simple Brownian Diffusion: An Introduction to the Standard Theoretical Models

*Daniel Thomas Gillespie and Effrosyni Seitaridou*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199664504
- eISBN:
- 9780191748516
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199664504.001.0001
- Subject:
- Physics, Soft Matter / Biological Physics

Brownian diffusion is the motion of one or more solute molecules in a sea of very many, much smaller solvent molecules. Its importance today owes mainly to cellular chemistry, since Brownian ... More

## A Multifactor, Nonlinear, Continuous‐Time Model of Interest Rate Volatility *

*Jacob Boudoukh, Christopher Downing, Matthew Richardson, Richard Stanton, and Robert F. Whitelaw*

### in Volatility and Time Series Econometrics: Essays in Honor of Robert Engle

- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199549498
- eISBN:
- 9780191720567
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199549498.003.0014
- Subject:
- Economics and Finance, Econometrics

This chapter provides a method for estimating multifactor continuous-time Markov processes. Using Milshtein's (1978) approximation schemes for writing expectations of functions of the sample path of ... More