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**Unformatted text preview: **Stochastic Processes Amir Dembo (revised by Kevin Ross) April 8, 2008 E-mail address : [email protected] Department of Statistics, Stanford University, Stanford, CA 94305. Contents Preface 5 Chapter 1. Probability, measure and integration 7 1.1. Probability spaces and σ-fields 7 1.2. Random variables and their expectation 10 1.3. Convergence of random variables 19 1.4. Independence, weak convergence and uniform integrability 25 Chapter 2. Conditional expectation and Hilbert spaces 35 2.1. Conditional expectation: existence and uniqueness 35 2.2. Hilbert spaces 39 2.3. Properties of the conditional expectation 43 2.4. Regular conditional probability 46 Chapter 3. Stochastic Processes: general theory 49 3.1. Definition, distribution and versions 49 3.2. Characteristic functions, Gaussian variables and processes 55 3.3. Sample path continuity 62 Chapter 4. Martingales and stopping times 67 4.1. Discrete time martingales and filtrations 67 4.2. Continuous time martingales and right continuous filtrations 73 4.3. Stopping times and the optional stopping theorem 76 4.4. Martingale representations and inequalities 82 4.5. Martingale convergence theorems 88 4.6. Branching processes: extinction probabilities 90 Chapter 5. The Brownian motion 95 5.1. Brownian motion: definition and construction 95 5.2. The reflection principle and Brownian hitting times 101 5.3. Smoothness and variation of the Brownian sample path 104 Chapter 6. Markov, Poisson and Jump processes 111 6.1. Markov chains and processes 111 6.2. Poisson process, Exponential inter-arrivals and order statistics 119 6.3. Markov jump processes, compound Poisson processes 125 Bibliography 127 Index 129 3 Preface These are the lecture notes for a one quarter graduate course in Stochastic Pro- cesses that I taught at Stanford University in 2002 and 2003. This course is intended for incoming master students in Stanford’s Financial Mathematics program, for ad- vanced undergraduates majoring in mathematics and for graduate students from Engineering, Economics, Statistics or the Business school. One purpose of this text is to prepare students to a rigorous study of Stochastic Differential Equations. More broadly, its goal is to help the reader understand the basic concepts of measure the- ory that are relevant to the mathematical theory of probability and how they apply to the rigorous construction of the most fundamental classes of stochastic processes. Towards this goal, we introduce in Chapter 1 the relevant elements from measure and integration theory, namely, the probability space and the σ-fields of events in it, random variables viewed as measurable functions, their expectation as the corresponding Lebesgue integral, independence, distribution and various notions of convergence. This is supplemented in Chapter 2 by the study of the conditional expectation, viewed as a random variable defined via the theory of orthogonal projections in Hilbert spaces....

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