Stochastic Processes

Courant Lecture Notes in Mathematics

Executive Editor Jalal Shatah

Managing Editor Paul D. Monsour

Assistant Editor Reeva Goldsmith

S. R. S. Varadhan Courant Institute of Mathematical Sciences

16 Stochasti c Processes

Courant Institute of Mathematical Science s New York University New York, New York

American Mathematical Societ y Providence, Rhode Island

http://dx.doi.org/10.1090/cln/016

2000 Mathematics Subject Classification. P r i m a r y 60G05 , 60G07 .

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Varadhan, S . R . S . Stochastic processe s / S . R . S . Varadhan .

p. cm . — (Couran t lectur e note s ; 16 ) Includes bibliographica l reference s an d index . ISBN 978-0-8218-4085- 6 (alk . paper ) 1. Stochasti c processes . I . Title .

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Dedication

To Gopal

I had planned to complete this book within a short time of the publication of the volume on probability theory . Bu t the events of September 11 , 2001, intervened. We lost our son Gopal that day, a victim of violence in the name of God. I dedicate this volume to his memory.

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Contents

Preface

Chapter 1 . Introductio n 1.1. Continuou s Time Processes 1.2. Continuou s Parameter Martingales 1.3. Semimartingale s 1.4. Martingale s and Stochastic Integrals

Chapter 2. Processe s with Independent Increments 2.1. Th e Basic Poisson Process 2.2. Compoun d Poisson Processes 2.3. Infinit e Number of Small Jumps 2.4. Infinitesima l Generator s 2.5. Som e Associated Martingales

Chapter 3. Poisso n Point Processes 3.1. Poin t Processes 3.2. Poisso n Point Process

Chapter 4. Jum p Markov Processes 4.1. Simpl e Examples 4.2. Semigroup s of Operators 4.3. Example : Birth and Death Processes 4.4. Marko v Processes and Martingales 4.5. Explosio n 4.6. Recurrenc e and Transience 4.7. Invarian t Distributions 4.8. Beyon d Explosion

Chapter 5. Brownia n Motion 5.1. Definitio n o f Brownian Motion 5.2. Marko v and Strong Markov Property 5.3. Hea t Equation 5.4. Recurrenc e 5.5. Feynman-Ka c Formula 5.6. Arcsin e Law 5.7. Harmoni c Oscillato r 5.8. Exi t Times from Bounded Intervals

ix

1 1 3 8

10

13 13 16 17 20 21

25 25 26

29 29 31 34 35 39 44 45 47

49 49 51 53 55 56 57 59 60

vii

viii CONTENT S

5.9. Stochasti c Integrals 5.10. Brownia n Motion with a Drift, Girsano v Fo 5.11. Ornstein-Uhlenbec k Proces s 5.12. Invarian t Densities 5.13. Loca l Times 5.14. Reflecte d Brownian Motion 5.15. Excursio n Theory 5.16. Invarianc e Principle 5.17. Representatio n of Martingales

Chapter 6. One-Dimensiona l Diffusion s 6.1. Stochasti c Differential Equation s 6.2. Propertie s of the Solution 6.3. Connection s with Differential Equation s 6.4. Martingal e Characterizatio n 6.5. Rando m Time Change 6.6. Som e Examples

Chapter 7. Genera l Theory of Markov Processes 7.1. Introductio n 7.2. Semigroups , Generators and Resolvents 7.3. Generator s and Martingales 7.4. Invarian t Measures and Ergodic Theory

Appendix A. Measure s on Polish Spaces A.l. Th e Space C[0, 1] A.2. Th e Space D[0, 1]

Appendix B. Additiona l Remarks

Bibliography

Index

Preface

This i s a continuation o f the volume o n probability theor y an d likewise cov-ers the contents of courses given at the Courant Institute. Thi s volume deals with certain elementary continuous-tim e processes . W e start with a description o f the Poisson process and related processes with independent increments . Afte r a brief look at Markov processes with a finite number of jumps we proceed to study Brow-nian motion. W e then go on to develop stochastic integrals and Ito's theory in the context of one-dimensional diffusion processes . I t ends with a brief surve y of the general theory of Markov processes.

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Bibliography

[1] Chung , K. L. Markov chains with stationary transition probabilities. 2nd ed. Die Grundlehren der mathematischen Wissenschaften, 104 . Springer, New York, 1967.

[2] Durrett , R . Stochastic calculus. A practical introduction. Probability an d Stochastic s Series . CRC Press, Boca Raton, Fla., 1996.

[3] Dynkin , E. B. Markov processes and semi-groups of operators. Teor. Veroyatnost. i Primenen. 1 (1956) , 25-37.

[4] Dynkin , E. B. One-dimensional continuous strong Markov processes. Theor. Probability Appl. 4 (1959), 1-52 .

[5] Parthasarathy , K. R. Probability measures on metric spaces. Reprint of the 1967 original. AMS Chelsea, Providence, R.I., 2005.

[6] Stroock , D. W.; Varadhan, S . R. S. Multidimensional diffusion processes. Reprin t of the 199 7 edition. Classics in Mathematics. Springer, Berlin, 2006.

[7] Varadhan , S . R. S . Probability theory. Couran t Lectur e Note s i n Mathematics , 7 . New Yor k University, Couran t Institut e o f Mathematica l Sciences , Ne w York ; America n Mathematica l Society, Providence, R.I., 2001.

[8] Wiener , N. Differential space . J. Math. Phys. 2 (1923), 132-174.

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Index

C[a, b], 3 D[a, b], 3

arcsine law, 57

Bessel process, 104 birth and death process, 34 Brownian motion, 49

geometric, 100 Markov property, 51 strong Markov property, 51

Chapman-Kolmogorov equations , 29 continuous-parameter martingale, 3

differential equation s and Markov processes, 94

Doob decomposition, 8 Doob's h-transform, 10 5 Doob's inequality, 4 Doob-Meyer decomposition, 8 Dynkin's formula, 11 0

excursion theory, 81 exit distribution, 36 exit time, 36

distribution, 60 explosion, 39, 71, 102

Feller's test, 102 Feynman-Kac formula, 56 filtration, 3

generator, 31 Girsanov formula, 69

harmonic oscillator, 59 heat equation, 53

infinitesimal generator , 20, 31 invariant distribution, 45, 75, 111

Ito's formula, 66 for stochastic integrals, 91

jump Markov process, 29 and strong Markov property, 32

Levy-Khintchine representation, 1 8 life after death, 47 local time, 76

martingale, 3, 22 exponential martingale, 23, 68 martingale problem, 97

one-dimensional diffusions, 8 7 option pricing, 101 optional stopping, 5, 6 Ornstein-Uhlenbeck process, 72 outer measure, 2

point process, 25 marked, 27 Poisson, 26

Poisson process, 13 compound, 1 6 rate, 15

Poisson random measure, 26 processes with independent increments, 17

quadratic variation, 61

random time change, 99 recurrence, 55 reflected Brownian motion, 79 reflection principle, 52 regularity

C[0, 1] , 116 D[0, 1] , 118

semigroup, 20, 31 semimartingale, 8 stable laws, 19

126

stochastic differential equatio n existence of, 8 7 properties of solutions of, 9 0 uniqueness of, 8 7

stochastic integration, 10 , 61 stochastic process, 1 stopped field, 4 stopping time, 4 submartingale, 4 supermartingale, 4

Tulceas' theorem, 30

Wiener's stochastic integral, 62