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Random walk approximation of Brownian motion. Tree methods. Computationally simple trees: the case of a constant diffusion and the general case. How to parallelize the algorithm.
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A general weak convergence result for diffusion processes. Monte Carlo simulation of a diffusion process: the Euler scheme. Discretization error and Monte Carlo error. How to parallelize the…
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Brownian motion. Nondifferentiability of paths. The stochastic integral and the Ito isometry. Ito processes and Ito formula. Stochastic differential equations.
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Markov chains: definition and examples. Simulation of a Markov chain. Simulation of Markov chains in the ergodic case.
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Variance reduction methods in Monte Carlo: antithetic variables, control variates, importance sampling. Examples and implementation
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Introduction to the course. Monte Carlo methods. Simulation of random variables. Error estimation with Monte Carlo methods.
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