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Part I (Voss, Lythe and Molina-Paris)
1-1 Discrete random variables. Continuous random variables. Independence.
1-2 Random walk. Binomial distribution, Poisson distribution.
1-3 Gambler's ruin. Occupation and exit time. First-step analysis.
1-4 Markov Chains. Birth and death processes. Gillespie algorithm.
1-5 Chapman-Kolmogorov equation (To glue to Markov chains?)
1-6 Stationary distributions, quasi-limiting distributions.
1-7 Stochastic processes. Wiener process. Diffusion equation.
1-8 The reflection principle and passage times. Conditional hitting probability.
1-9 Ornstein-Uhlenbeck processes. Bessel processes.
1-10 Numerical methods.
1-11 Applications to biology.
Part II (Veretennikov) Syllabus
2-1 Stochastic processes; some measure theory; Kolmogorov continuity theorem.
2-2 Filtrations and conditional expectations.
2-3 Wiener measure.
2-4 Stochastic Ito integrals.
2-5 Stopping times; martingales; Kolmogorov and Doob theorems.
2-6 Ito formula.
2-7 Stochastic differential equations, existence and uniqueness of solutions.
2-8 Passage times, links to Laplace and Poisson equations;Dynkin and Feynman-Kac formulae.
2-9 Girsanov change of measure; weak solutions of SDEs.
2-10 Dependence of solutions of SDEs from initial data; Markov property of solutions.