Probability and Stochastics

This text is an introduction to the modern theory and applications of probability and stochastics. The style and coverage is geared towards the theory of stochastic processes, but with some attention to the applications. In many instances the gist of the problem is introduced in practical, everyday language and then is made precise in mathematical form. The first four chapters ...

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This text is an introduction to the modern theory and applications of probability and stochastics. The style and coverage is geared towards the theory of stochastic processes, but with some attention to the applications. In many instances the gist of the problem is introduced in practical, everyday language and then is made precise in mathematical form. The first four chapters are on probability theory: measure and integration, probability spaces, conditional expectations, and the classical limit theorems. There follows chapters on martingales, Poisson random measures, Levy Processes, Brownian motion, and Markov Processes. Special attention is paid to Poisson random measures and their roles in regulating the excursions of Brownian motion and the jumps of Levy and Markov processes. Each chapter has a large number of varied examples and exercises. The book is based on the author's lecture notes in courses offered over the years at Princeton University. These courses attracted graduate students from engineering, economics, physics, computer sciences, and mathematics. Erhan Cinlar has received many awards for excellence in teaching, including the President's Award for Distinguished Teaching at Princeton University. His research interests include theories of Markov processes, point processes, stochastic calculus, and stochastic flows. The book is full of insights and observations that only a lifetime researcher in probability can have, all told in a lucid yet precise style.

Preface v
Frequently Used Notation ix
I Measure and Integration 1
1 Measurable Spaces . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . 6
3 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
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Preface v
Frequently Used Notation ix
I Measure and Integration 1
1 Measurable Spaces . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . 6
3 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 Transforms and Indefinite Integrals . . . . . . . . . . . . . . . 29
6 Kernels and Product Spaces . . . . . . . . . . . . . . . . . . . 37
II Probability Spaces 49
1 Probability Spaces and Random Variables . . . . . . . . . . . 50
2 Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Lp-spaces and Uniform Integrability . . . . . . . . . . . . . . 70
4 Information and Determinability . . . . . . . . . . . . . . . . 75
5 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
III Convergence 93
1 Convergence of Real Sequences . . . . . . . . . . . . . . . . . 93
2 Almost Sure Convergence . . . . . . . . . . . . . . . . . . . . 97
3 Convergence in Probability . . . . . . . . . . . . . . . . . . . 101
4 Convergence in Lp . . . . . . . . . . . . . . . . . . . . . . . . 105
5 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . 109
6 Laws of Large Numbers . . . . . . . . . . . . . . . . . . . . . 118
7 Convergence of Series . . . . . . . . . . . . . . . . . . . . . . 124
8 Central Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 127
IV Conditioning 139
1 Conditional Expectations . . . . . . . . . . . . . . . . . . . . 139
2 Conditional Probabilities and Distributions . . . . . . . . . . 149
3 Conditional Independence . . . . . . . . . . . . . . . . . . . . 158
4 Construction of Probability Spaces . . . . . . . . . . . . . . . 160
5 Special Constructions . . . . . . . . . . . . . . . . . . . . . . 166
V Martingales and Stochastics 171
1 Filtrations and Stopping Times . . . . . . . . . . . . . . . . . 171
2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3 Martingale Transformations and Maxima . . . . . . . . . . . 190
4 Martingale Convergence . . . . . . . . . . . . . . . . . . . . . 199
5 Martingales in Continuous Time . . . . . . . . . . . . . . . . 213
6 Martingale Characterizations for Wiener and Poisson . . . . . 225
7 Standard Filtrations and Modifications of Martingales . . . . 234
VI Poisson Random Measures 243
1 Random Measures . . . . . . . . . . . . . . . . . . . . . . . . 243
2 Poisson Random Measures . . . . . . . . . . . . . . . . . . . . 248
3 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 263
4 Additive Random Measures and L´evy Processes . . . . . . . . 277
5 Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . 290
6 Poisson Integrals and Self-exciting Processes . . . . . . . . . . 298
VII L´evy Processes 313
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
2 Stable Processes . . . . . . . . . . . . . . . . . . . . . . . . . 329
3 L´evy Processes on Standard Settings . . . . . . . . . . . . . . 340
4 Characterizations for Wiener and Poisson . . . . . . . . . . . 349
5 Itˆo-L´evy Decomposition . . . . . . . . . . . . . . . . . . . . . 354
6 Subordination . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
7 Increasing L´evy Processes . . . . . . . . . . . . . . . . . . . . 368
VIII Brownian Motion 379
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
2 Hitting Times and Recurrence Times . . . . . . . . . . . . . . 389
3 Hitting Times and Running Maximum . . . . . . . . . . . . . 396
4 Wiener and its Maximum . . . . . . . . . . . . . . . . . . . . 399
5 Zeros, Local Times . . . . . . . . . . . . . . . . . . . . . . . . 408
6 Excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
7 Path Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 426
8 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
IX Markov Processes 443
1 Markov Property . . . . . . . . . . . . . . . . . . . . . . . . . 444
2 Itˆo Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
3 Jump-Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . 473
4 Markov Systems . . . . . . . . . . . . . . . . . . . . . . . . . 498
5 Hunt Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 505
6 Potentials and Excessive Functions . . . . . . . . . . . . . . . 518
7 Appendix: Stochastic Integration . . . . . . . . . . . . . . . . 525
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