Applied Stochastic Processes presents a concise, graduate-level treatment of the subject, emphasizing applications and practical computation. It also establishes the complete mathematical theory in an accessible way. After reviewing basic probability, the text covers Poisson processes, renewal processes, discrete- and continuous-time Markov chains, and Brownian motion. It also offers an introduction to stochastic differential equations. While the main applications described are queues, the book also considers other examples, such as the mathematical model of a single stock market. With exercises in most sections, this book provides a clear, practical introduction for beginning graduate students. The material is presented in a straightforward manner using short, motivating examples. In addition, the author develops the mathematical theory with a strong emphasis on probability intuition
Probability and Stochastic Processes Probability Random variables and their distributions Mathematical expectation Joint distribution and independence Convergence of random variables Laplace transform and generating functions Examples of discrete distributions Examples of continuous distributions Stochastic processes Stopping times Conditional expectation Poisson Processes Introduction to Poisson processes Arrival and inter-arrival times of Poisson processes Conditional distribution of arrival times Poisson processes with different types of events Compound Poisson processes Nonhomogeneous Poisson processes Renewal Processes An introduction to renewal processes Renewal reward processes Queuing systems Queue lengths, waiting times, and busy periods Renewal equation Key renewal theorem Regenerative processes Queue length distribution and PASTA Discrete Time Markov Chains Markov property and transition probabilities Examples of discrete time Markov chains Multi-step transition and reaching probabilities Classes, recurrence, and transience Periodicity, class property, and positive recurrence Expected hitting time and hitting probability Stationary distribution Limiting properties Continuous Time Markov Chain Markov property and transition probability Transition rates Stationary distribution and limiting properties Birth and death processes Exponential queuing systems Time reversibility Hitting time and phase-type distributions Queuing systems with time-varying rates Brownian Motion and Beyond Brownian motion Standard Brownian motion and its maximum Conditional expectation and martingales Brownian motion with drift Stochastic integrals Ito's formula and stochastic differential equations A single stock market model Bibliography Index
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