Multidimensional Diffusion Processes

by Stroock, Daniel W.; Varadhan, S. R. S.

• ISBN13: 9783540289982
• ISBN10: 3540289984
• Edition: Reprint
• Format: Paperback
• Copyright: 2006-01-15
• Publisher: Springer Verlag

Summary

"This book is an excellent presentation of the application of martingale theory to the theory of Markov processes, especially multidimensional diffusions. This approach was initiated by Stroock and Varadhan in their famous papers. (...) The proofs and techniques are presented in such a way that an adaptation in other contexts can be easily done. (...) The reader must be familiar with standard probability theory and measure theory which are summarized at the beginning of the book. This monograph can be recommended to graduate students and research workers but also to all interested in Markov processes from a more theoretical point of view." Mathematische Operationsforschung und Statistik, 1981

Author Biography

Daniel W. Stroock is Simons Professor of Mathematics at the Massachusetts Institute of Technology in Cambridge, Massachusetts, USA.

Table of Contents

Frequently Used Notation	xi
Chapter 0. Introduction	1	(6)
Chapter 1. Preliminary Material: Extension Theorems, Martingales, and Compactness	7	(39)
1.0 Introduction	7	(1)
1.1 Weak Convergence, Conditional Probability Distributions and Extension Theorems	7	(12)
1.2 Martingales	19	(11)
1.3 The Space C([0, infinity); Rd)	30	(6)
1.4 Martingales and Compactness	36	(6)
1.5 Exercises	42	(4)
Chapter 2. Markov Processes, Regularity of Their Sample Paths, and the Wiener Measure	46	(19)
2.1 Regularity of Paths	46	(5)
2.2 Markov Processes and Transition Probabilities	51	(5)
2.3 Wiener Measure	56	(4)
2.4 Exercises	60	(5)
Chapter 3. Parabolic Partial Differential Equations	65	(17)
3.1 The Maximum Principle	65	(6)
3.2 Existence Theorems	71	(8)
3.3 Exercises	79	(3)
Chapter 4. The Stochastic Calculus of Diffusion Theory	82	(40)
4.1 Brownian Motion	82	(3)
4.2 Equivalence of Certain Martingales	85	(7)
4.3 Itô Processes and Stochastic Integration	92	(12)
4.4 Itô's Formula	104	(3)
4.5 Itô Processes as Stochastic Integrals	107	(4)
4.6 Exercises	111	(11)
Chapter 5. Stochastic Differential Equations	122	(14)
5.0 Introduction	122	(2)
5.1 Existence and Uniqueness	124	(7)
5.2 On the Lipschitz Condition	131	(1)
5.3 Equivalence of Different Choices of the Square Root	132	(2)
5.4 Exercises	134	(2)
Chapter 6. The Martingale Formulation	136	(35)
6.0 Introduction	136	(3)
6.1 Existence	139	(6)
6.2 Uniqueness: Markov Property	145	(4)
6.3 Uniqueness: Some Examples	149	(3)
6.4 Cameron-Martin-Girsanov Formula	152	(5)
6.5 Uniqueness: Random Time Change	157	(4)
6.6 Uniqueness: Localization	161	(4)
6.7 Exercises	165	(6)
Chapter 7. Uniqueness	171	(24)
7.0 Introduction	171	(3)
7.1 Uniqueness: Local Case	174	(13)
7.2 Uniqueness: Global Case	187	(3)
7.3 Exercises	190	(5)
Chapter 8. Itô's Uniqueness and Uniqueness to the Martingale Problem	195	(13)
8.0 Introduction	195	(1)
8.1 Results of Yamada and Watanabe	195	(9)
8.2 More on Itô Uniqueness	204	(3)
8.3 Exercises	207	(1)
Chapter 9. Some Estimates on the Transition Probability Functions	208	(40)
9.0 Introduction	208	(1)
9.1 The Inhomogeneous Case	209	(24)
9.2 The Homogeneous Case	233	(15)
Chapter 10. Explosion	248	(13)
10.0 Introduction	248	(1)
10.1 Locally Bounded Coefficients	249	(5)
10.2 Conditions for Explosion and Non-Explosion	254	(5)
10.3 Exercises	259	(2)
Chapter 11. Limit Theorems	261	(24)
11.0 Introduction	261	(1)
11.1 Convergence of Diffusion Process	262	(4)
11.2 Convergence of Markov Chains to Diffusions	266	(6)
11.3 Convergence of Diffusion Processes: Elliptic Case	272	(7)
11.4 Convergence of Transition Probability Densities	279	(4)
11.5 Exercises	283	(2)
Chapter 12. The Non-Unique Case	285	(19)
12.0 Introduction	285	(1)
12.1 Existence of Measurable Choices	286	(4)
12.2 Markov Selections	290	(6)
12.3 Reconstruction of All Solutions	296	(6)
12.4 Exercises	302	(2)
Appendix	304	(24)
A.0 Introduction	304	(2)
A.1 Lp Estimates for Some Singular Integral Operators	306	(9)
A.2 Proof of the Main Estimate	315	(8)
A.3 Exercises	323	(5)
Bibliographical Remarks	328	(3)
Bibliography	331	(6)
Index	337