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. Preface ... 7 Contents ... 9 Review of Probability ... 11 1.1 Events and Probability ... 11 1.2 Random Variables ... 13 1.3 Conditional Probability and Independence ... 18 1.4 Solutions ... 20 Conditional Expectation ... 26 2.1 Conditioning on an Event ... 26 2.2 Conditioning on a Discrete Random Variable ... 28 2.3 Conditioning on an Arbitrary Random Variable ... 31 2.4 Conditioning on a ?-Field ... 36 2.5 General Properties ... 38 2.6 Various Exercises on Conditional Expectation ... 40 2.7 Solutions ... 42 Martingales in Discrete Time ... 53 3.1 Sequences of Random Variables ... 53 3.2 Filtrations ... 54 3.3 Martingales ... 56 3.4 Games of Chance ... 59 3.5 Stopping Times ... 62 3.6 Optional Stopping Theorem ... 66 3.7 Solutions ... 69 Martingale Inequalities and Convergence ... 74 4.1 Doob's Martingale Inequalities ... 75 4.2 Doob's Martingale Convergence Theorem ... 78 4.3 Uniform Integrability and L1 Convergence of Martingales ... 80 4.4 Solutions ... 87 Markov Chains ... 91 5.1 First Examples and Definitions ... 92 5.2 Classification of States ... 107 5.3 Long-Time Behaviour of Markov Chains: General Case ... 114 5.4 Long-Time Behaviour of Markov Chains with Finite State Space ... 120 5.5 Solutions ... 125 Stochastic Processes in Continuous Time ... 144 6.1 General Notions ... 144 6.2 Poisson Process ... 145 6.2.1 Exponential Distribution and Lack of Memory ... 145 6.2.2 Construction of the Poisson Process ... 147 6.2.3 Poisson Process Starts from Scratch at Time t ... 150 6.2.4 Various Exercises on the Poisson Process ... 153 6.3 Brownian Motion ... 155 6.3.1 Definition and Basic Properties ... 156 6.3.2 Increments of Brownian Motion ... 158 6.3.4 Doob's Maximal L2 Inequality for Brownian Motion ... 164 6.4 Solutions ... 166 Ito Stochastic Calculus ... 183 7.1 Ito Stochastic Integral: Definition ... 184 7.2 Examples ... 193 7.3 Properties of the Stochastic Integral ... 194 7.4 Stochastic Differential and Ito Formula ... 197 7.5 Stochastic Differential Equations ... 206 7.6 Solutions ... 213 Index ... 227 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