## Stochastic ModelingThree coherent parts form the material covered in this text, portions of which have not been widely covered in traditional textbooks. In this coverage the reader is quickly introduced to several different topics enriched with 175 exercises which focus on real-world problems. Exercises range from the classics of probability theory to more exotic research-oriented problems based on numerical simulations. Intended for graduate students in mathematics and applied sciences, the text provides the tools and training needed to write and use programs for research purposes. The first part of the text begins with a brief review of measure theory and revisits the main concepts of probability theory, from random variables to the standard limit theorems. The second part covers traditional material on stochastic processes, including martingales, discrete-time Markov chains, Poisson processes, and continuous-time Markov chains. The theory developed is illustrated by a variety of examples surrounding applications such as the gambler’s ruin chain, branching processes, symmetric random walks, and queueing systems. The third, more research-oriented part of the text, discusses special stochastic processes of interest in physics, biology, and sociology. Additional emphasis is placed on minimal models that have been used historically to develop new mathematical techniques in the field of stochastic processes: the logistic growth process, the Wright –Fisher model, Kingman’s coalescent, percolation models, the contact process, and the voter model. Further treatment of the material explains how these special processes are connected to each other from a modeling perspective as well as their simulation capabilities in C and MatlabTM. |

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Assume birth and death bond percolation chapter chosen uniformly communication classes conditional expectation contact process continuous-time Markov chain convergence theorem converges almost surely critical value death process deduce define discrete-time Markov chain edges Example Exercise expected value exponentially distributed fraction gambler’s ruin chain gives graph implies infinite integer lattice interacting particle systems irreducible jumps keeps track large numbers law of large Lemma Let Xn limn logistic growth process martingale Moran model number of individuals offspring optional stopping theorem parameter particular player Poisson point process Poisson processes positive recurrent printf probability theory process starting prove random walk result sample paths sequence shows simple birth simulate site percolation space spatial stationary distribution step stochastic processes subgraph symmetric random walk transient transition probabilities transition rates type 1 individuals uniformly at random uniformly integrable vertex vertices voter model Wright–Fisher model zero