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E-grāmata: Stochastic Analysis and Diffusion Processes

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(Professor Emeritus, Department of Statistics, University of North Carolina at Chapel Hill), (Professor of Mathematics, Department of Mathematics, Louisiana State University)
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Stochastic Analysis and Diffusion ProcessesPalielināt
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Stochastic Analysis and Diffusion Processes presents a simple, mathematical introduction to Stochastic Calculus and its applications. The book builds the basic theory and offers a careful account of important research directions in Stochastic Analysis. The breadth and power of Stochastic Analysis, and probabilistic behavior of diffusion processes are told without compromising on the mathematical details.

Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. The book proceeds to construct stochastic integrals, establish the Itō formula, and discuss its applications. Next, attention is focused on stochastic differential equations (SDEs) which arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of SDEs and form the main theme of this book.

The Stroock-Varadhan martingale problem, the connection between diffusion processes and partial differential equations, Gaussian solutions of SDEs, and Markov processes with jumps are presented in successive chapters. The book culminates with a careful treatment of important research topics such as invariant measures, ergodic behavior, and large deviation principle for diffusions.

Examples are given throughout the book to illustrate concepts and results. In addition, exercises are given at the end of each chapter that will help the reader to understand the concepts better. The book is written for graduate students, young researchers and applied scientists who are interested in stochastic processes and their applications. The reader is assumed to be familiar with probability theory at graduate level. The book can be used as a text for a graduate course on Stochastic Analysis.

Recenzijas

Very readable * Paul Taylor, Mathematics Today * If I were giving a graduate course on this topic, then I would certainly use this book. * Dave Applebaum, The Mathematical Gazette * The book can be recommended for all specialists in probability and stochastic processes and its applications starting from the undergraduate and graduate students and ending with experienced professionals. * Yuliya S. Mishura, Zentralblatt MATH *

1 Introduction to Stochastic Processes
1(18)
1.1 The Kolmogorov Consistency Theorem
1(10)
1.2 The Language of Stochastic Processes
11(3)
1.3 Sigma Fields, Measurability, and Stopping Times
14(5)
Exercises
17(2)
2 Brownian Motion
19(22)
2.1 Definition and Construction of Brownian Motion
20(7)
2.2 Essential Features of a Brownian Motion
27(7)
2.3 The Reflection Principle
34(7)
Exercises
39(2)
3 Elements of Martingale Theory
41(34)
3.1 Definition and Examples of Martingales
41(3)
3.2 Wiener Martingales and the Markov Property
44(5)
3.3 Essential Results on Martingales
49(5)
3.4 The Doob-Meyer Decomposition
54(13)
3.5 The Meyer Process for L2-martingales
67(4)
3.6 Local Martingales
71(4)
Exercises
73(2)
4 Analytical Tools for Brownian Motion
75(15)
4.1 Introduction
75(1)
4.2 The Brownian Semigroup
76(3)
4.3 Resolvents and Generators
79(8)
4.4 Pregenerators and Martingales
87(3)
Exercises
89(1)
5 Stochastic Integration
90(44)
5.1 The Ito Integral
90(8)
5.2 Properties of the Integral
98(7)
5.3 Vector-valued Processes
105(1)
5.4 The Ito Formula
106(5)
5.5 An Extension of the Ito Formula
111(2)
5.6 Applications of the Ito Formula
113(11)
5.7 The Girsanov Theorem
124(10)
Exercises
132(2)
6 Stochastic Differential Equations
134(32)
6.1 Introduction
134(3)
6.2 Existence and Uniqueness of Solutions
137(7)
6.3 Linear Stochastic Differential Equations
144(2)
6.4 Weak Solutions
146(7)
6.5 Markov Property
153(8)
6.6 Generators and Diffusion Processes
161(5)
Exercises
164(2)
7 The Martingale Problem
166(36)
7.1 Introduction
166(8)
7.2 Existence of Solutions
174(9)
7.3 Analytical Tools
183(6)
7.4 Uniqueness of Solutions
189(4)
7.5 Markov Property of Solutions
193(3)
7.6 Further Results on Uniqueness
196(6)
8 Probability Theory and Partial Differential Equations
202(38)
8.1 The Dirichlet Problem
202(10)
8.2 Boundary Regularity
212(6)
8.3 Kolmogorov Equations: The Heuristics
218(3)
8.4 Feynman-Kac Formula
221(2)
8.5 An Application to Finance Theory
223(1)
8.6 Kolmogorov Equations
224(16)
Exercises
239(1)
9 Gaussian Solutions
240(26)
9.1 Introduction
241(4)
9.2 Hilbert-Schmidt Operators
245(3)
9.3 The Gohberg-Krein Factorization
248(4)
9.4 Nonanticipative Representations
252(5)
9.5 Gaussian Solutions of Stochastic Equations
257(9)
Exercises
265(1)
10 Jump Markov Processes
266(26)
10.1 Definitions and Basic Results
266(5)
10.2 Stochastic Calculus for Processes with Jumps
271(4)
10.3 Jump Markov Processes
275(8)
10.4 Diffusion Approximation
283(9)
Exercises
290(2)
11 Invariant Measures and Ergodicity
292(23)
11.1 Introduction
293(2)
11.2 Ergodicity for One-dimensional Diffusions
295(6)
11.3 Invariant Measures for d-dimensional Diffusions
301(3)
11.4 Existence and Uniqueness of Invariant Measures
304(6)
11.5 Ergodic Measures
310(5)
Exercises
314(1)
12 Large Deviations Principle for Diffusions
315(28)
12.1 Definitions and Basic Results
316(2)
12.2 Large Deviations and Laplace-Varadhan Principle
318(11)
12.3 A Variational Representation Theorem
329(9)
12.4 Sufficient Conditions for LDP
338(5)
Exercises
341(2)
Notes on
Chapters		 343(4)
References 347(4)
Index 351
Gopinath Kallianpur, Professor Emeritus at University of North Carolina at Chapel Hill, has worked extensively on Stochastic Analysis and is a world renowned expert on stochastic filtering theory. He is the author of Stochastic Filtering Theory, and a co-author of White Noise Theory of Prediction, Filtering and Smoothing, Introduction to Option Pricing Theory, and Stochastic Differential Equations in Infinite Dimensions.



P. Sundar is a Professor of Mathematics at Louisiana State University. He works on Stochastic Analysis, and is on the Editorial Board for the journal Communications on Stochastic Analysis. He has co-edited a book titled Infinite Dimensional Stochastic Analysis.
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