Basic Stochastic Processes : A Course Through Exercises

Stochastic processes are tools used widely by statisticians and researchers working in the mathematics of finance. This book for self-study provides a detailed treatment of conditional expectation and probability, a topic that in principle belongs to probability theory, but is essential as a tool for stochastic processes. The book centers on exercises as the main means of explanation. Title......Page 4 Copyright Page......Page 5 Preface......Page 8 Contents......Page 10 1.1 Events and Probability......Page 12 1.2 Random Variables......Page 14 1.3 Conditional Probability and Independence......Page 19 1.4 Solutions......Page 21 2.1 Conditioning on an Event......Page 28 2.2 Conditioning on a Discrete Random Variable......Page 30 2.3 Conditioning on an Arbitrary Random Variable......Page 33 2.4 Conditioning on a a-Field......Page 38 2.5 General Properties......Page 40 2.6 Various Exercises on Conditional Expectation......Page 42 2.7 Solutions......Page 44 3.1 Sequences of Random Variables......Page 56 3.2 Filt rations......Page 57 3.3 Martingales......Page 59 3.4 Games of Chance......Page 62 3.5 Stopping Times......Page 65 3.6 Optional Stopping Theorem......Page 69 3.7 Solutions......Page 72 4. Martingale Inequalities and Convergence......Page 78 4.1 Doob 's Martingale Inequalities......Page 79 4.2 Doob's Martingale Convergence Theorem......Page 82 4.3 Uniform Integrability and ?1 Convergence of Martingales......Page 84 4.4 Solutions......Page 91 5. Markov Chains......Page 96 5.1 First Examples and Definitions......Page 97 5.2 Classification of States......Page 112 5.3 Long-Time Behaviour of Markov Chains: General Case......Page 119 5.4 Long-Time Behaviour of Markov Chains with Finite State Space......Page 125 5.5 Solutions......Page 130 6.1 General Notions......Page 150 6.2.1 Exponential Distribution and Lack of Memory......Page 151 6.2.2 Construction of the Poisson Process......Page 153 6.2.3 Poisson Process Starts from Scratch at Time t......Page 156 6.2.4 Various Exercises on the Poisson Process......Page 159 6.3 Brownian Motion......Page 161 6.3.1 Definition and Basic Properties......Page 162 6.3.2 Increments of Brownian Motion......Page 164 6.3.3 Sample Paths......Page 167 6.3.4 Doob's Maximal L2 Inequality for Brownian Motion......Page 170 6.3.5 Various Exercises on Brownian Motion......Page 171 6.4 Solutions......Page 172 7. Ito Stochastic Calculus......Page 190 7.1 Ito Stochastic Integral: Definition......Page 191 7.2 Examples......Page 200 7.3 Properties of the Stochastic Integral......Page 201 7.4 Stochastic Differential and Ito Formula......Page 204 7.5 Stochastic Differential Equations......Page 213 7.6 Solutions......Page 220 Index......Page 234 This book has been designed for a final year undergraduate course in stochastic processes. It will also be suitable for mathematics undergraduates and others with interest in probability and stochastic processes, who wish to study on their own. The main prerequisite is probability theory: probability measures, random variables, expectation, independence, conditional probability, and the laws of large numbers. The only other prerequisite is calculus. This covers limits, series, the notion of continuity, differentiation and the Riemann integral. Familiarity with the Lebesgue integral would be a bonus. A certain level of fundamental mathematical experience, such as elementary set theory, is assumed implicitly. Throughout the book the exposition is interlaced with numerous exercises, which form an integral part of the course. Complete solutions are provided at the end of each chapter. Also, each exercise is accompanied by a hint to guide the reader in an informal manner. This feature willbe particularly useful for self-study and may be of help in tutorials. It also presents a challenge for the lecturer to involve the students as active participants in the course. 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. Preliminaries -- Stochastic Processes: Case Studies -- Markov Chains -- Spectral Theory Of Stationary Processes -- Renewal Theory -- Martingales -- Itô Stochastic Processes. Zdzisław Brzeźniak And Tomasz Zastawniak. Includes Bibliographical References And Index. "This book is a final year undergraduate text on stochastic processes, a tool used widely by statisticians and researchers working, for example, in the mathematics of finance. The book will give a detailed treatment of conditional expectation and probability, a topic which is essential as a tool for stochastic processes. Although the book is a final year text, the authors have chosen to use exercises as the main means of explanation for the various topics, hence the course has a strong self-study element. The authors have concentrated on major topics within stochastic analysis: martingales in discrete time and their convergence, Markov chains, stochastic processes in continuous time, with emphasis on the Poisson process and Brownian motion, as well as Ito stochastic calculus including stochastic differential equations."--Jacket