Optimal Stopping Rules

by Shiryaev, A. N.; Aries, A. B.

• ISBN13: 9783540740100
• ISBN10: 3540740104
• Edition: Reprint
• Format: Paperback
• Copyright: 2007-12-01
• Publisher: Springer Nature

Summary

Although three decades have passed since first publication of this book reprinted now as a result of popular demand, the content remains up-to-date and interesting for many researchers as is shown by the many references to it in current publications.

Table of Contents

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
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