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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.

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…

Brownian motion. Nondifferentiability of paths. The stochastic integral and the Ito isometry. Ito processes and Ito formula. Stochastic differential equations.

Markov chains: definition and examples. Simulation of a Markov chain. Simulation of Markov chains in the ergodic case.

Variance reduction methods in Monte Carlo: antithetic variables, control variates, importance sampling. Examples and implementation

Introduction to the course. Monte Carlo methods. Simulation of random variables. Error estimation with Monte Carlo methods.

End of Lectures 1-2

Lectures 1-2 (sudden fall of Internet connection at the end)