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Tables of Contents for Handbook of Stochastic Analysis and Applications
Chapter/Section Title
Page #
Page Count
Preface
iii
Contributors
xvii
Markov Processes and Their Applications
1
46
Rabi Bhattacharya
Introduction
1
3
Markov Chains
4
7
Simple Random Walk
5
1
Birth-Death Chains and the Ehrenfest Model
6
1
Galton-Watson Branching Process
7
1
Markov Chains in Continuous Time
8
3
References
11
1
Discrete Parameter Markov Processes on General State Spaces
11
9
Ergodicity of Harris Recurrent Processes
11
3
Iteration of I.I.D. Random Maps
14
3
Ergodicity of Non-Harris Processes
17
2
References
19
1
Continuous Time Markov Processes on General State Spaces
20
2
Processes with Independent Increments
20
1
Jump Processes
21
1
References
22
1
Markov Processes and Semingroup Theory
22
9
The Hille-Yosida Theorem
23
2
Semigroups and One-Dimensional Diffusions
25
6
References
31
1
Stochastic Differential Equations
31
16
Stochastic Integrals, SDE, Ito's Lemma
32
3
Cameron-Martin-Girsanov Theorem and the Martingale Problem
35
3
Probabilistic Representation of Solutions to Elliptic and Parabolic Partial Differential Equations
38
1
References
39
2
Bibliography
41
6
Semimartingale Theory and Stochastic Calculus
47
60
Jia-An Yan
General Theory of Stochastic Processes and Martingale Theory
48
20
Classical Theory of Martingales
48
4
General Theory of Stochastic Processes
52
8
Modern Martingale Theory
60
8
Stochastic Integrals
68
19
Stochastic Integrals w.r.t. Local Martingales
68
4
Stochastic Integrals w.r.t. Semimartingales
72
3
Convergence Theorems for Stochastic Integrals
75
3
Ito's Formula and Doleans Exponential Formula
78
3
Local Times of Semimartingales
81
1
Fisk-Stratonovich Integrals
82
2
Stochastic Differential Equations
84
3
Stochastic Calculus on Semimartingales
87
20
Stochastic Integration w.r.t. Random Measures
87
3
Characteristics of a Semimartingale
90
1
Processes with Independent Increments and Levy Processes
91
3
Absolutely Continuous Changes of Probability
94
5
Martingale Representation Theorems
99
4
Bibliography
103
4
White Noise Theory
107
52
Hui-Hsuing Kuo
Introduction
107
4
What is white noise?
107
1
White noise as the derivative of a Brownian motion
107
1
The use of white noise--a simple example
108
1
White noise as a generalized stochastic process
109
1
White noise as an infinite dimensional generalized function
110
1
White noise as a distribution theory
111
6
Finite dimensional Schwartz distribution theory
111
1
White noise space
112
1
Hida's original idea
112
2
Spaces of test and generalized functions
114
1
Examples of test and generalized functions
115
2
General spaces of test and generalized functions
117
6
Abstract white noise space
117
1
Wick tensors
118
1
Hida-Kubo-Takenaka space
119
1
Kondratiev-Streit space
120
1
Cochran-Kuo-Sengupta space
121
2
Continuous versions and analytic extensions
123
8
Continuous versions
123
2
Analytic extensions
125
1
Integrable functions
126
2
Generalized functions induced by measures
128
1
Generalized Radon-Nikodym derivative
129
2
Characterization theorems
131
9
The S-transform
131
1
Characterization of generalized functions
132
4
Convergence of generalized functions
136
1
Characterization of test functions
137
2
Intrinsic topology for the space of test functions
139
1
Continuous operators and adjoints
140
10
Differential operators
140
3
Translation and scaling operators
143
1
Multiplication and Wick product
144
2
Fourier-Gauss transform
146
2
Extensions to CKS-spaces
148
2
Comments on other topics and applications
150
9
Bibliography
155
4
SDEs and Their Applications
159
78
Bo Zhang
SDEs with respect to Brownian motion
160
9
Ito type SDEs
160
3
Properties of solutions
163
2
Equations depending on a parameter
165
2
Stratonovich Stochastic Differential Equations
167
1
Stochastic Differential Equations on Manifolds
168
1
Applications
169
22
Diffusions
169
4
Boundary value problem
173
3
Optimal stopping
176
4
Stochastic control
180
5
Backward SDE and applications
185
6
Some generalizations of SDEs
191
21
SDEs of the jump type
191
7
SDE with respect to semimartingale
198
6
SDE driven by nonlinear integrator
204
8
Stochastic Functional Differential Equations
212
7
Existence and Uniqueness of Solution
212
3
Markov property
215
2
Regularity of the trajectory field
217
2
Stochastic Differential Equations in Abstract Spaces
219
5
Stochastic evolution equations
219
3
Dissipative stochastic systems
222
2
Anticipating Stochastic Differential Equation
224
13
Volterra equations with anticipating kernel
224
3
SDEs with anticipating drift and initial condition
227
2
Bibliography
229
8
Numerical Analysis of SDEs Without Tears
237
124
H. Schurz
Introduction
237
1
The Standard Setting For (O)SDEs
238
3
Stochastic Taylor Expansions
241
5
The Ito Formula (Ito's Lemma)
241
1
The main idea of stochastic Ito's-Taylor expansions
241
2
Hierarchical sets, coeffcient functions, multiple integrals
243
1
Amore compact formulation
243
1
The example of Geometric Brownian Motion
244
1
Key relations between multiple integrals
245
1
A Toolbox of Numerical Methods
246
19
The explicit and fully drift-implicit Euler method
246
1
The family of stochastic Theta methods
247
1
Trapezoidal and midpoint methods
248
1
Rosenbrock methods (RTMs)
248
1
Balanced implicit methods (BIMs)
249
1
Predictor-corrector methods (PCMs)
249
1
Explicit Runge-Kutta methods (RKMs)
250
1
Newton's method
251
1
The explicit and implicit Mil'shtein methods
252
1
Gaines's representation of Mil'shtein method
253
1
Generalized Theta-Platen methods
254
1
Talay-Tubaro extrapolation technique and linear PDEs
254
1
Denk-Hersch method for highly oscillating systems
255
2
Stochastic Adams-type methods
257
1
The two step Mil'shtein method of Horvath-Bokor
258
1
Higher order Taylor methods
258
1
Splitting methods of Petersen-Schurz
258
2
The ODE method with commutative noise
260
2
Random local linearization methods (LLM)
262
2
Simultaneous time and chance discretizations
264
1
Stochastic waveform relaxation methods
264
1
Comments on numerical analysis of SPDEs
264
1
General concluding comment on numerical methods
265
1
On the Main Principles of Numerics
265
11
ID-invariance
265
1
Numerical pth mean consistency
266
1
Numerical pth mean stability
266
1
Numerical pth mean contractivity
267
1
Numerical pth mean convergence
267
1
The main principle: combining all concepts from 5.1-5.5
268
5
On fundamental crossrelations
273
3
Results on Convergence Analysis
276
15
Continuous time convergence concepts
276
2
On key relations between convergence concepts
278
1
Fundamental theorems of mean square convergence
278
2
Strong mean square convergence theorem
280
1
The Clark-Cameron mean square order bound in IR1
280
2
Exact mean square order bounds of Cambanis and Hu
282
2
Atheorem on double L2-convergence with adaptive Δn
284
1
The fundamental theorem of weak convergence
285
1
Approximation of some functionals
286
3
The pathwise error process for explicit Euler methods
289
1
Almost sure convergence
289
2
Numerical Stability, Stationarity, Boundedness, and Invariance
291
18
Stability of linear systems with ultiplicative noise
291
3
Stationarity of linear systems with additive noise
294
2
Asymptotically exact methods for linear systems
296
1
Almost sure nonnegativity of numerical methods
297
2
Numerical invariance of intervals [0, M]
299
2
Preservation of bounaries for Brownian Bridges
301
1
Nonlinear stability of implict Euler methods
302
1
Linear and nonlinear A-stability
303
1
Stability exponents of explicit-implicit methods
304
2
Hofmann-Platen's M-stability concept in C1
306
2
Asymptotic stability with probability one
308
1
Numerical Contractivity
309
8
Contractivity of SDEs with monotone coefficients
309
1
Contractivity of implicit Euler methods
310
1
pth mean B- and BN-stability
310
1
Contractivity exponents of explicit-implicit methods
311
1
General V-asymptotics of discrete time iterations
312
2
An example for discrete time V-asymptotics
314
3
Asymptotic contractivity with probability one
317
1
On Practical Implementation
317
13
Implementation issues: some challenging examples
317
4
Generation of pseudorandom numbers
321
2
Substitutions of randomness under weak convergence
323
1
Are quasi random numbers useful for (O)SDEs
324
1
Variable step size algorithms
325
1
Variance reduction techniques
326
2
How to estimate pth mean errors
328
1
On software and programmed packages
329
1
Comments on applications of numerics for (O)SDEs
329
1
Comments, Outlook, Further Developments
330
31
Recent and further developments
330
1
General comments
330
1
Acknowledgements
331
1
New trends-10 challenging problem areas
331
2
Bibliography
333
28
Large Deviations and Applications
361
56
Amir Dembo
Ofer Zeitouni
Introduction
361
2
The Large Deviation Principle
363
2
Large Deviation Principles for Finite Dimensional Spaces
365
8
The Method of Types
366
2
Cramer's Theorem in IRd
368
1
The Gartner-Ellis Theorem
369
2
Inequalities for Bounded Martingale Differnces
371
1
Moderate Deviations and Exact Asymptotics
372
1
General Properties
373
15
Existence of an LDP and Related Properties
374
2
Contraction Principles and Exponential Approximation
376
4
Varadhan's Lemma and its Converse
380
2
Convexity Considerations
382
3
Large Deviations for Projective Limits
385
3
Sample Path LDPs
388
8
Sample Path Large Deviations for Random Walk and for Brownian Motion
388
2
The Freidlin-Wentzell Theory
390
2
Application: The Problem of Diffusion Exit from a Domain
392
4
LDPs for Empirical Measures
396
21
Cramer's Theorem in Polish Spaces
396
3
Sanov's Theorem
399
2
LDP for Empirical Measures of Markov Chains
401
3
Mixing Conditions and LDP
404
2
Application: The Gibbs Conditioning Principle
406
4
Application: The Hypothesis Testing Problem
410
3
Bibliography
413
4
Stability and Stabilizing Control of Stochastic Systems
417
56
P. V. Pakshin
Stochastic mathematical models of systems
419
6
Models of differential systems corrupted by noise
419
3
Models of differential systems with random jumps
422
1
Differential generator
423
2
Stochastic control problem
425
13
Preliminaries
425
1
Stochastic dynamic programming
426
4
Stochastic maximum principle
430
2
Separation principle
432
6
Definition of stochastic stability and stochastic Lyapunov function
438
4
Classic stability concept
438
1
Weak Lyapunov stability
438
1
Strong Lyapunov stability
439
1
Mean square and p-stability
439
1
Recurrence and positivity
440
1
Stochastic Lyapunov function
441
1
General stability and stabilization theorems
442
6
Stability in probability theorems
442
1
Recurrence and positivity theorems
442
1
pth mean stability theorems and their inversion
443
3
Stability in the first order approximation
446
1
Stabilization problem and fundamental theorem
447
1
Instability
448
3
Classic stochastic instability concept
448
2
Nonpositivity and nonrecurrence
450
1
Stability criteria and testable conditions
451
7
General stability tests for linear systems
451
1
Some particular stability criteria for linear systems
452
2
Stability of the pth moments of linear systems
454
1
Absolute stochastic stability
455
1
Robust stability
456
2
Stabilizing control of linear system
458
15
General linear systems
458
1
Linear systems with parametric noise
459
5
Robust stabilizing control
464
3
Bibliography
467
6
Stochastic Differential Games and Applications
473
60
K. M. Ramachandran
Introduction
473
2
Two person zero-sum differential games
475
15
Two person zero-sum games: martingale methods
475
9
Two person zero-sum games and viscosity solutions
484
3
Stochastic differential games with multiple modes
487
3
N-Person stochastic differential games
490
8
Discounted payoff on the infinite horizon
491
1
Ergodic payoff
492
6
Weak convergence methods in differential games
498
20
Weak convergence preliminaries
498
2
Weak convergence in N-person stochastic differential games
500
10
Partially observed stochastic differential games and weak convergence
510
8
Applications
518
5
Stochastic equity investment model with institutional investor speculation
519
4
Conclusion
523
10
Bibliography
525
8
Stochastic Manufacturing Systems: A Hierarchial Control Approach
533
44
Q. Zhang
Introduction
533
2
Single Machine System
535
3
Flowshops
538
3
Jobshops
541
1
Production-Capacity Expansion Models
542
6
Production-Marketing Models
548
2
Risk-Sensitive Control
550
3
Optimal Control
553
2
Hierarchical Control
555
2
Risk-Sensitive Control
557
3
Constant Product Demand
560
6
Constant Machine Capacity
566
2
Marketing-Production with a Jump Demand
568
3
Concluding Remarks
571
6
Bibliography
573
4
Stochastic Approximation: Theory and Applications
577
48
G. Yin
Introduction
577
2
Historical Development
578
1
Basic Issues
579
1
Outline of the Chapter
579
1
Algorithms and Variants
579
6
Basic Algorithm
579
2
More General Algorithms
581
1
Projection and Truncation Algorithms
582
2
Global Stochastic Approximation
584
1
Continuous-time Stochastic Approximation Algorithms
585
1
Stochastic Approximation in Function Spaces
585
1
Convergence
585
5
ODE Methods
586
2
Weak Convergence Method
588
2
Rates of Convergence
590
4
Scaling Factor α
590
1
Tightness of the Scaled Estimation Error
591
1
Local Analysis
592
2
Random Directions
594
1
Stopping Rules
594
1
Large Deviations
594
2
Motivation
595
1
Large Deviations for Stochastic Approximation
595
1
Asymptotic Efficiency
596
5
Iterate Averaging
597
1
Smoothed Algorithms
598
2
Some Numerical Data
600
1
Applications
601
13
Adaptive Filtering
602
1
Adaptive Beam Forming
602
1
System Identification and Adaptive Control
603
2
Adaptive Step-size Tracking Algorithms
605
1
Approximation of Threshold Control Policies
606
1
GI/G/1 Queue
607
1
Distributed Algorithms for Supervised Learning
608
2
A Heat Exchanger
610
2
Evolutionary Algorithms
612
1
Digital Diffusion Machines
613
1
Further Remarks
614
11
Convergence
614
1
Rate of Convergence
615
1
Law of Iterated Logarithms
615
1
Robustness
616
1
Parallel Stochastic Approximation
616
1
Open Questions
617
1
Conclusion
617
2
Bibliography
619
6
Optimization by Stochastic Methods
625
54
Franklin Mendivil
R. Shonkwiler
M. C. Spruill
Nature of the problem
625
4
Introduction
625
1
No Free Lunch
626
2
The Permanent Problem
628
1
A Brief Survey of Some Methods for Global Optimization
629
6
Covering Methods
630
1
Branch and bound
631
1
Iterative Improvement
632
1
Trajectory/tunneling Methods
633
1
Tabu search
634
1
Random Search
634
1
Multistart
635
1
Markov Chain and Renewal Theory Considerations
635
9
IIP parallel search
638
1
Restarted Improvement Algorithms
639
3
Renewal Techniques in Restarting
642
2
Simulated Annealing
644
9
Introduction
644
2
Simulated annealing applied to the permanent problem
646
1
Convergence Properties of Simulated Annealing and Related Algorithms
647
6
Restarted Algorithms
653
5
Introduction
653
1
The Permanent Problem using restarted simulated annealing
654
1
Restarted Simulated Annealing
655
1
Numerical comparisons
656
2
Evolutionary Computations
658
21
Introduction
658
2
A GA for the permanent problem
660
1
Some specific Algorithms
661
1
GA principles, schemata, multi-armed bandit, implicit parallelism
662
5
A genetic algorithm for constrained optimization problems
667
3
Markov Chain Analysis Particular to Genetic Algorithms
670
3
Bibliography
673
6
Stochastic Control Methods in Asset Pricing
679
75
Thaleia Zariphopoulou
Introduction
679
1
The Hamilton-Jacobi-Bellman (HJB) equation
680
4
Models of Optimal Investment and Consumption I
684
23
Merton models with intermediate consumption
687
2
Merton models with non-linear stock dynamics
689
2
Merton models with trading constraints
691
2
Merton models with non-homogeneous investment opportunities
693
6
Models of Optimal Portfolio Management with General Utilities
699
4
Optimal goal problems
703
2
Alternative models of expected utility
705
2
Models of optimal investment and consumption II
707
16
Optimal investment/consumption models with transaction costs
707
12
Optimal investment/consumption models with stochastic labor income
719
4
Expected utility methods in derivative pricing
723
31
The Black and Scholes valuation formula
725
2
Super-replicating strategies
727
2
The utility maximization theory
729
9
Imperfect hedging strategies
738
4
Other models of derivative pricing with transaction costs
742
3
Bibliography
745
9
Index
754