Basic Stochastic Processes

Description

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