Stochastic Modelling and Applied Probability

By: A.B. Aries (Translator), Albert N. Shiryaev

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Paperback | 7 November 2007 | Edition Number 3

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236 Pages

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Revised

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22.86 x 15.24 x 1.27

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Along with conventional problems of statistics and probability, the - vestigation of problems occurring in what is now referred to as stochastic theory of optimal control also started in the 1940s and 1950s. One of the most advanced aspects of this theory is the theory of optimal stopping rules, the development of which was considerably stimulated by A. Wald, whose Sequential ~nal~sis' was published in 1947. In contrast to the classical methods of mathematical statistics, according to which the number of observations is fixed in advance, the methods of sequential analysis are characterized by the fact that the time at which the observations are terminated (stopping time) is random and is defined by the observer based on the data observed. A. Wald showed the advantage of sequential methods in the problem of testing (from independent obser- tions) two simple hypotheses. He proved that such methods yield on the average a smaller number of observations than any other method using fixed sample size (and the same probabilities of wrong decisions). Furth- more, Wald described a specific sequential procedure based on his sequ- tial probability ratio criterion which proved to be optimal in the class of all sequential methods. By the sequential method, as applied to the problem of testing two simple hypotheses, we mean a rule according to which the time at which the observations are terminated is prescribed as well as the terminal decision as to which of the two hypotheses is true.

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You Can Find This Book In

Non-Fiction Mathematics Probability & Statistics Applied Mathematics Stochastics Economics Econometrics Economic Statistics

Random Processes: Markov Times	p. 1
Background material from the theory of probability	p. 1
Markov times	p. 5
Martingales and semimartingales	p. 15
Markov processes	p. 18
Notes to Chapter 1	p. 24
Optimal Stopping of Markov Sequences	p. 25
Statement of the problem of optimal stopping	p. 25
Optimal stopping rules in the classes [characters not reproducible](n) and [characters not reproducible](m; n)	p. 28
An optimal selection problem	p. 35
Excessive functions and smallest excessive majorants	p. 39
The excessive characterization of the payoff and [epsilon]-optimal stopping rules (under the condition A[superscript -])	p. 50
Examples	p. 57
The structure and methods of finding a payoff for a function g [set membership] B (a[superscript -])	p. 64
Regular functions: the structure of the payoff and [epsilon]-optimal stopping rules (under the condition A[superscript +])	p. 68
Regular characterization of the payoff (the general case)	p. 75
Convergence of the payoffs s[subscript n](x) and the optimal times [tau superscript * subscript n] as n to [infinity]	p. 76
Solutions of recursive equations f(x) = max{g(x, Tf(x)}	p. 79
Criteria for the truncation of optimal stopping rules	p. 86
Randomized and sufficient classes of stopping times	p. 90
Optimal stopping of a Markov sequence allowing for the cost of observation	p. 93
Reduction of the optimal stopping problem for arbitrary random sequences to the corresponding problem for Markov processes	p. 100
Notes to Chapter 2	p. 111
Optimal Stopping of Markov Processes	p. 113
The statement of the problem and main definitions	p. 113
Regular and excessive functions: excessive majorants	p. 115
Excessive characterization of the payoff and [epsilon]-optimal stopping times (under the condition A[superscript -])	p. 124
Regular characterization of the payoff and [epsilon]-optimal stopping times (under the condition A[superscript +])	p. 129
Regular characterization of the payoff (the general case)	p. 138
The construction of regular majorants	p. 142
[epsilon](x)-optimal Markov times	p. 153
Equations for the payoff and generalized Stefan problem: the conditions for "smooth pasting"	p. 157
Notes to Chapter 3	p. 162
Some Applications to Problems of Mathematical Statistics	p. 163
The sequential testing of two simple hypotheses (discrete time)	p. 163
Sequential testing of two simple hypotheses on the mean of a Wiener process	p. 180
The problem of disruption (discrete time)	p. 193
The problem of disruption for a Wiener process	p. 200
Notes to Chapter 4	p. 207
Bibliography	p. 208
Index	p. 215
Table of Contents provided by Ingram. All Rights Reserved.

Stochastic Modelling and Applied Probability

At a Glance

Paperback

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How to return your order

Defective items

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