Simulation and Monte Carlo Methods

Objective

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.

Planning

1. 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
2. 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.
3. 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.

References

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