Basic Stochastic Processes

by Brzezniak, Zdzisaw; Zastawniak, Tomasz

• ISBN13: 9783540761754
• ISBN10: 3540761756
• Format: Paperback
• Copyright: 1998-10-01
• Publisher: Springer Verlag

Summary

This book is a final year undergraduate text on stochastic processes, a tool used widely by statisticians and researchers working in the mathematics of finance. The book will give a detailed treatment of conditional expectation and probability, a topic which in principle belongs to probability theory, but is essential as a tool for stochastic processes. Although the book is a final year text, the author has chosen to use exercises as the main means of explanation for the various topics, and the book will have a strong self-study element. The author has concentrated on the major topics within stochastic analysis: Stochastic Processes, Markov Chains, Spectral Theory, Renewal Theory, Martingales and Itô Stochastic Processes.

Table of Contents

1. Review of Probability	1	(16)
1.1 Events and Probability	1	(2)
1.2 Random Variables	3	(5)
1.3 Conditional Probability and Independence	8	(2)
1.4 Solutions	10	(7)
2. Conditional Expectation	17	(28)
2.1 Conditioning on an Event	17	(2)
2.2 Conditioning on a Discrete Random Variable	19	(3)
2.3 Conditioning on an Arbitrary Random Variable	22	(5)
2.4 Conditioning on a Alpha-Field	27	(2)
2.5 General Properties	29	(2)
2.6 Various Exercises on Conditional Expectation	31	(2)
2.7 Solutions	33	(12)
3. Martingales in Discrete Time	45	(22)
3.1 Sequences of Random Variables	45	(1)
3.2 Filtrations	46	(2)
3.3 Martingales	48	(3)
3.4 Games of Chance	51	(3)
3.5 Stopping Times	54	(4)
3.6 Optional Stopping Theorem	58	(3)
3.7 Solutions	61	(6)
4. Martingale Inequalities and Convergence	67	(18)
4.1 Doob's Martingale Inequalities	68	(3)
4.2 Doob's Martingale Convergence Theorem	71	(2)
4.3 Uniform Integrability and L(1) Convergence of Martingales	73	(7)
4.4 Solutions	80	(5)
5. Markov Chains	85	(54)
5.1 First Examples and Definitions	86	(15)
5.2 Classification of States	101	(7)
5.3 Long-Time Behaviour of Markov Chains: General Case	108	(6)
5.4 Long-Time Behaviour of Markov Chains with Finite State Space	114	(5)
5.5 Solutions	119	(20)
6. Stochastic Processes in Continuous Time	139	(40)
6.1 General Notions	139	(1)
6.2 Poisson Process	140	(10)
6.2.1 Exponential Distribution and Lack of Memory	140	(2)
6.2.2 Construction of the Poisson Process	142	(3)
6.2.3 Poisson Process Starts from Scratch at Time t	145	(3)
6.2.4 Various Exercises on the Poisson Process	148	(2)
6.3 Brownian Motion	150	(11)
6.3.1 Definition and Basic Properties	151	(2)
6.3.2 Increments of Brownian Motion	153	(3)
6.3.3 Sample Paths	156	(3)
6.3.4 Doob's Maximal L(2) Inequality for Brownian Motion	159	(1)
6.3.5 Various Exercises on Brownian Motion	160	(1)
6.4 Solutions	161	(18)
7. Ito Stochastic Calculus	179	(44)
7.1 Ito Stochastic Integral: Definition	180	(9)
7.2 Examples	189	(1)
7.3 Properties of the Stochastic Intergral	190	(3)
7.4 Stochastic Differential and Ito Formula	193	(9)
7.5 Stochastic Differential Equations	202	(7)
7.6 Solutions	209	(14)
Index	223