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-field a.s. continuous paths adapted belongs Borel function Borel set called Chapman-Kolmogorov equations completing the proof compute conditional expectation conditional probability convergence theorem countable cr-field defined Definition 2.3 denote differential equation dX(t dW(t equality Exercise exists exponential distribution filtration Tn follows gambling strategy Hence Hint implies increments induction inequality integrable random variable invariant measure Ito formula Ito process joint density Lemma M2tep Markov chain Markov property martingale with respect mn(j non-negative null-recurrent Optional Stopping Theorem partition Poisson process positive-recurrent probability measure probability space Proposition proves random step processes random walk real numbers recurrent right-hand side sample path satisfies sequence of random Show Solution stochastic differential equation stochastic integral stochastic matrix stochastic process submartingale supermartingale Suppose Theorem 7.1 total probability formula transient uniformly integrable unique invariant measure upcrossings verify Wiener process