Introduction to stochastic processes
著者
書誌事項
Introduction to stochastic processes
(Chapman & Hall probability series)
Chapman & Hall, c1995
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注記
Includes index
内容説明・目次
内容説明
This concise, informal introduction to stochastic processes evolving with time was designed to meet the needs of graduate students not only in mathematics and statistics, but in the many fields in which the concepts presented are important, including computer science, economics, business, biological science, psychology, and engineering.
With emphasis on fundamental mathematical ideas rather than proofs or detailed applications, the treatment introduces the following topics:
Markov chains, with focus on the relationship between the convergence to equilibrium and the size of the eigenvalues of the stochastic matrix
Infinite state space, including the ideas of transience, null recurrence and positive recurrence
The three main types of continual time Markov chains and optimal stopping of Markov chains
Martingales, including conditional expectation, the optional sampling theorem, and the martingale convergence theorem
Renewal process and reversible Markov chains
Brownian motion, both multidimensional and one-dimensional
Introduction to Stochastic Processes is ideal for a first course in stochastic processes without measure theory, requiring only a calculus-based undergraduate probability course and a course in linear algebra.
目次
PRELIMINARIES
Introduction
Linear Differential Equations
Linear Difference Equations
FINITE MARKOV CHAINS
Definitions and Examples
Long-Range Behavior and Invariant Probability
Classification of States
Return Times
Transient States
Examples
COUNTABLE MARKOV CHAINS
Introduction
Recurrence and Transience
Positive Recurrence and Null Recurrence
Branching Process
CONTINUOUS-TIME MARKOV CHAINS
Poisson Process
Finite State Space
Birth-and-Death Processes
General Case
OPTIMAL STOPPING
Optimal Stopping of Markov Chains
Optimal Stopping with Cost
Optimal Stopping with Discounting
MARTINGALES
Conditional Expectation
Definition and Examples
Optional Sampling theorem
Uniform Integrability
Martingale Convergence Theorem
RENEWAL PROCESSES
Introduction
Renewal Equation
Discrete Renewal Processes
M/G/1 and B/M/1 Queues
REVERSIBLE MARKOV CHAINS
Reversible Processes
Convergence to Equilibrium
Markov Chain Algorithms
A Criterion for Recurrence
BROWNIAN MOTION
Introduction
Markov Property
Zero Set of Brownian Motion
Brownian Motion in Several Dimensions
Recurrence and Transience
Fractal Nature of Brownian Motion
Brownian Motion with Drift
STOCHASTIC INTEGRATION
Integration with Respect to Random walk
Integration with Respect to Brownian Motion
Ito's Formula
Simulation
INDEX
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