The aim of this course is to study the foundations of the so-called Monte Carlo and quasi-Monte Carlo methods, which are widely used in particular for the numerical valuation of financial and insurance products.
- General remarks– Reminders about the convergence of moments estimators. Uniform law generators. Other law generators: distribution function inversion method, rejection method and conditional laws, transformation method with application to the generation of Gaussians, correlated variables and mixture of laws and conditioning approach
- Error control and the variance reduction method– Reminders of techniques for evaluating estimation error. Antithetic control. Control variable. Importance sampling. Stratification and post-stratification. Latin hypercube sampling.
- Quasi-Monte Carlo method –Uniform sequences over the unit cube and discrepancy. Functions of limited variation in the sense of measures, Koksma-Hlawka inequality and numerical integration. Example of low-discrepancy sequences. Randomized deterministic sequences. Concepts of effective dimension and reduction of total variation.
DEVROYE Luc, Non-Uniform Random Variate Generation, Springer
GLASSERMAN Paul, Monte Carlo Methods in Financial Engineering
LEMIEUX Christiane, Monte Carlo and quasi-Monte Carlo sampling, Springer
ROBERT Christian P., Monte Carlo statistical methods, Springer
TUFFIN Bruno, La simulation de Monte Carlo, Hermes Science