It is desirable to know something about Markov chains.
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Fridays
Lythe (weeks 1-4)
Voss (weeks 5-8)
Molina-Paris (weeks 9-10)
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1-1 Gambler's ruin. Discrete random variables. Continuous random variables.
1-2 Random walk, discrete-time Markov chains.
1-3 Branching processes. Continuous-time Markov Chains. Birth and death processes. Gillespie algorithm.
1-4 Stationary distributions, quasi-limiting distributions.
1-5 Stochastic processes. Wiener process. Diffusion equation.
1-6 The reflection principle and passage times. Conditional hitting probability.
1-9 Applications to immunology.
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Mondays (Veretennikov)
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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.