Comparison methods for stochastic models and risks / Alfred Müller, Dietrich Stoyan.

Format:

Book

Author/Creator:

Mueller, Alfred.

Contributor:

Stoyan, Dietrich.

Anne and Joseph Trachtman Memorial Book Fund.

Series:

Wiley series in probability and statistics

Language:

English

Subjects (All):

Stochastic systems.

Physical Description:

xii, 330 pages : illustrations ; 23 cm.

Place of Publication:

Chichester ; New York : Wiley, 2002.

Summary:

Stochastic order relations prprovide a valuable insight into the behaviour of complex stochastic (random) systems and enable the user to collect meaningful comparative data. Application areas include queueing systems, actuarial and financial risk, decision making and stochastic simulation. Applicable to a broad range of scientific disciplines, including economics, finance, insurance and operations research Provides coverage of the latest research and applications An essential resource for researchers and postgraduate students appliying stochastic order relations, and scientisits from applied statistics, operations research, economics and finance.

Contents:

1 Univariate Stochastic Orders 1

1.2 Usual Stochastic Order 2

1.3 Hazard Rate Order 8

1.4 Likelihood Ratio Order 12

1.5 Convex Orders 15

1.5.1 Fundamental Properties 15

1.5.2 Sufficient Conditions and Strassen's Theorem 23

1.5.3 Majorization 31

1.6 Higher Convexity Orders and Laplace Transform Order 37

1.7 Dispersive Order and Relative Inverse Function Orderings 40

1.8 Lifetime Distributions and Notions of Aging 45

1.9 Bivariate Characterizations 51

1.10 Extremal Elements 55

1.11 Monotone Approximations 59

1.12 Relationships and Comparison Criteria for Univariate Stochastic Orders 60

2 Theory of Integral Stochastic Orders 65

2.2 Tools from Functional Analysis 67

2.3 Maximal Generators of Integral Stochastic Orders 69

2.4 Properties of Stochastic Orders 73

2.5 Small Generators 75

2.6 Strassen Type Theorems 80

3 Multivariate Stochastic Orders 85

3.2 Properties of Multivariate Stochastic Orders 89

3.3 Usual Stochastic Order and Orthant Orders 90

3.4 Convex Orders 98

3.5 Linear Convex Orders 101

3.6 Componentwise Convex Order 103

3.7 Stochastic Orders Defined by Difference Operators 105

3.8 Dependence Orders 107

3.9 Supermodular Order 112

3.10 Concepts of Dependence 121

3.11 Multivariate Likelihood Ratio Orders 129

3.12 Directionally Convex Order 131

3.13 Stochastic Ordering of Multivariate Normal Distributions 141

3.14 Relationships and Comparison Criteria for Multivariate Stochastic Orders 145

4 Stochastic Models, Comparison and Monotonicity 149

4.1 General Considerations Concerning Stochastic Models 149

4.2 Monotonicity and Comparability 154

4.2.1 Monotonicity 154

4.2.2 Comparability 154

4.3 Methods for Establishing Monotonicity and Comparability Properties 155

4.3.1 The Functional Method 155

4.3.2 The Mapping Method 156

4.3.3 The Coupling Method 166

4.4 Extremal Problems 171

5 Monotonicity and Comparability of Stochastic Processes 173

5.2 Comparability and Monotonicity of Markov Processes 180

5.2.1 Monotone and Comparable Operators 180

5.2.2 Monotonicity and Comparability Conditions for Markov Processes 185

5.2.3 Homogeneous Markov Processes with Discrete State Space 192

5.2.4 Monotonicity Properties of Second Order Characteristics of Markov Chains 198

5.2.5 Application of Monotone Markov Chains: Perfect Simulation 201

5.2.6 Markov Decision Processes 204

5.3 Monotonicity and Comparability of Non-Markov Processes 207

5.4 Comparison of Point Processes 211

6 Monotonicity Properties and Bounds for Queueing Systems 217

6.1 Basic Facts for GI/GI/1 and G/G/1 217

6.2 Monotonicity Properties of GI/GI/1 and G/G/1 Queues 220

6.3 Comparison Properties of GI/GI/1 and G/G/1 221

6.4 Bounds Obtained from Comparison Properties of GI/GI/1 225

6.5 Bounds in the Case of Non-renewal Input 226

6.6 Basic Facts for the Multi-server System GI/GI/s 228

6.7 Monotonicity Properties of GI/GI/s Queues 229

6.8 Comparability Properties of GI/GI/s 231

6.9 Remarks on other Queueing Systems 235

7 Applications to Various Stochastic Models 237

7.1 Monotonicity Properties and Bounds for the Renewal Function 237

7.2 Reliability Applications 239

7.2.1 Coherent Systems 239

7.2.2 Comparison of Maintenance Policies 242

7.3 PERT and Scheduling Problems 244

7.4 Comparison of Random Sets and Point Processes 247

7.4.1 Comparison of Random Closed Sets 247

7.4.2 Comparison of Point Processes 251

7.5 Monotonicity and Comparison of Models of Statistical Physics 253

7.5.1 Monotonicity and Comparison Properties of the Ising Model 253

7.5.2 Comparison of Gibbs Distributions 259

8 Comparing Risks 265

8.1 Economics of Uncertainty 265

8.1.1 Basics of Stochastic Dominance 265

8.1.2 First- and Second-Order Stochastic Dominance 266

8.1.3 Stochastic Dominance with DARA Utility Functions 269

8.2 Financial Applications 274

8.2.1 Consistency of Mean-deviation Rules 274

8.2.2 Portfolio Optimization 275

8.3 Ordering of Actuarial Risks 278

8.3.1 Bounds for Aggregate Claims of Dependent Risks 278

8.3.2 Some Models for Dependent Risks 289

8.3.3 Indistinguishable Individuals 294

8.3.4 Distinguishable Individuals 296.

Notes:

Includes bibliographical references (pages [303]-323) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Anne and Joseph Trachtman Memorial Book Fund.

ISBN:

0471494461

OCLC:

48754108