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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.

Stochastische Ordnungsbeziehungen geben Aufschlusse uber das Verhalten komplexer stochastischer (zufalliger) Systeme und ermoglichen die Zusammenstellung signifikanter Vergleichsdaten. Zu den Anwendungsgebieten gehoren die Organisation von Warteschlangen-Systemen (Queueing), die Abschatzung von Versicherungs- und Finanzrisiken, die Entscheidungstheorie und die stochastische Simulation, eingesetzt in den verschiedensten Wissenschaftsbereichen (Okonomie, Finanzwirtschaft, Versicherungen, Operations-Research). Dieser Band faßt fur Sie die neueste Forschungsergebnisse auf diesem Gebiet und aktuelle Anwendungen stochastischer Modelle zusammen.

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Stochastic orders are important approximation tools that provide valuable insight into the behaviour of complex stochastic models. Research into stochastic orders is blossoming, with many open problems being studied and a wide range of applications explored. In this book the authors explore the most important concepts of the field, from the basic univariate theory through to the most current applications.
* Comprehensive coverage of the theory and applications of stochastic orders.

* Employs a systematic approach with detailed explanation of each concept.

* Features coverage of univariate and multivariate stochastic orders.

* Covers a range of applications, from queueing theory, reliability theory, statistical physics, actuarial and financial risk, and economics.

* Written by authors with many years experience in researching stochastic orders.
Researchers and graduate students studying stochastic orders will find the comprehensive coverage of the subject ideal for their needs. The range of applications will benefit those working in applied probability and statistics, reliability, actuarial science, economics and finance.

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CONTENTS
Preface = ⅸ
1. Univariate Stochastic Orders = 1
 1.1 Introduction = 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
 2.1 Introduction = 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
 3.1 Preliminaries = 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
 5.1 Introduction = 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
List of Symbols = 299
References = 303
Index = 325