Mass Transportation Problems. Vol. I: Theory. Coauthor: S. T. Rachev. Springer Verlag 1998, ca. 470 pg. (Language: english)
 
   
Contents to Volume I
 

Preface to Volume I
vii

Preface to Volume II
xv

1
Introduction
1
  1.1 Mass Transportation Problems in Probability Theory 1
  1.2 Specially Structured Uansportation Problems 21
  1.3 Two Examples of the Interplay Between Continuous
and Discrete MTPs
23
  1.4 Stochastic Applications 27

2
The Monge-Kantorovich Problem
57
  2.1 The Multivariate Monge-Kantorovich Problem:
An Introduction		 58
  2.2 Primal and Dual Monge-Kantorovich Functionals 64
  2.3 Duality Theorems in a Topological Setting 76
  2.4 General Duality Theorem 82
  2.5 Duality Theorems with Metric Cost Functions 86
  2.6 Dual Representation for Lp-Minimal Metrics 96

3
Explicit Results for the Monge-Kantorovich Problem
107
  3.1 The One-Dimensional Case 107
  3.2 The Convex Case 112
  3.3 The General Case 123
  3.4 An Extension of the Kantorovich L2-Minimal Problem 132
  3.5 Maximum Probability of Sets, Maximum of Sums,
and Stochastic Order
144
  3.6 Hoeffding-Fréchet Bounds 151
  3.7 Bounds for the Total lyansportation Cost 158

4
Duality Theory for Mass Transfer Problems
161
  4.1 Duality in the Compact Case 161
  4.2 Cost Functions with Triangle Inequality 172
  4.3 Reduction Theorems 190
  4.4 Proofs of the Main Duality Theorenis and a Discussion 207
  4.5 Duality Theorems for Noncompact Spaces 219
  4.6 Infinite Linear Programs 241
  4.6.1 Duality Theory for an Abstract Scheme
of Infinite-Dimensional Linear Programs
and Its Application to the Mass Transfer Problem		  
 
241
  4.6.2 Duality Theorems for the Mass Transfer Problem
with Given Marginals		  
245
  4.6.3 Duality Theorem for a Marginal Problem
with Additional Constraints of Moment-Type		  
251
  4.6.4 Duality theorem for a Further Extremal
Marginal Problem		  
258
  4.6.5 Duality Theorem for a Nontopological Version
of the Mass Transfer Problem		  
265
 
5		  
Applications of the Duality Theory		  
275
  5.1 Mass Transfer Problem with a Smooth
Cost Function-Explicit Solution		  
275
  5.2 Extension and Approximate Extension Theorems 290
  5.2.1 The Simplest Extension Theorem
(the Case X = E(S) and X1 = E(S1))		  
290
  5.2.2 Approximate Extension Theorems 292
  5.2.3 Extension Theorenis 295
  5.2.4 A continuous selection theorem 302
  5.3 Approximation Theorems  306
  5.4 An Application of the Duality Theory
to the Strassen Theorem		  
319
  5.5 Closed Preorders and Continuous Utility Functions 322
  5.5.1 Statement of the Problem and the Idea
of the Duality Approach		  
322
  5.5.2 Functionally Closed Preorders 324
  5.5.3 Two Generalizations of the Debreu Theorem 329
  5.5.4 The Case of a Locally Compact Space 335
  5.5.5 Varying preorders and a universal utility theorem 337
  5.5.6 Functionally Closed Preorders
and Strong Stochastic Dominance		  
341
  5.6 Further Applications to Utility Theory 344
  5.6.1 Preferences That Admit Lipschitz
or Continuous Utility Functions 		 
344
  5.6.2 Application to Choice Theory
in Mathematical Economics		  
352
  5.7 Applications to Set-Valued Dynamical Systems 354
  5.7.1 Compact-Valued Dynamical Systems:
Quasiperiodic Points 		 
354
  5.7.2 Compact-Valued Dynamical Systems:
Asymptotic Behavior of Trajectories		  
358
  5.7.3 A Dynamic Optimization Problem 363
  5.8 Compensatory Transfers and Action Profiles 367
 
6		  
Mass Transshipment Problems and Ideal Metrics 		 
371
  6.1 Kantorovich-Rubinstein Problems with Constraints 372
  6.2 Constraints on the k-th Difference of Marginals 383
  6.3 The General Case 402
  6.4 Minimality of Ideal Metrics 414
 
References		  
429
 
Abbreviations		  
473
 
Symbols		  
475
 
Index		  
478
  back last update   27. June 2001