Basic Stochastic Processes: A Course Through Exercises
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 will be 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.
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a.s. continuous paths adapted belongs Borel function Borel set called ChapmanKolmogorov equations completing the proof compute conditional expectation conditional probability convergence theorem countable defined Definition 2.3 denote distributed random variables equality Example Exercise exists exponential distribution finite followsthat forany gambling strategy Hence Hint implies increments independent identically distributed induction integrable random variables inthe invariant measure Itô formula Itô process Jensen’s inequality joint density Lemma Markov chain Markov property martingale martingale with respect nondecreasing nonnegative ofthe Poisson process positiverecurrent probability measure Proposition proves random step processes random walk randomvariable recurrent Riemann integral righthand side satisfies sequence of random Show square integrable random stochastic differential equation stochastic integral stochastic matrix stochastic process submartingale supermartingale Suppose thatthe Theorem 7.1 tosses uniformly integrable unique invariant measure upcrossings verify weshall Wiener process ξ τ