Section: New Results

Monte-Carlo methods

Adaptive variance reduction methods.

Stochastic algorithms [52] , [53] , [80] , [63] , [81] , [27] or, more recently, direct stochastic optimization [73] proved to be a promising path to automatic variance reduction methods. Direct stochastic optimization techniques are easier to use in practice, avoiding completely any manual tuning needed for stochastic algorithms. This method is well understood (see [73] ) only in the Gaussian case and for regular functions. We plan to extend the algorithms and prove rigorous results in non Gaussian cases with financial applications in view for jumps models (see [77] , [76] , [75] ).

Monte-Carlo methods for calibration.

The interest for models combining local and stochastic volatility has been growing recently. Indeed, the local volatility model is not rich enough to efficiently deal with complex derivatives. A popular model is the so called Heston model, in which the volatility process solves a square-root stochastic differential equation (just as in the Cox-Ingersoll-Ross model for interest rate modeling). The thesis of L. Abbas-Turki [12] (advisers: D. Lamberton and B. Lapeyre) concentrates on the multi-dimensional Heston model. For these models, numerical aspects are very demanding and we plan to use Monte-Carlo methods using advanced parallel devices (GPU clusters,...) both for price computations and calibration procedures. The thesis of Abbas-Turki is supported by the Pôle de Compétitivité Finance Innovation within the consortium CrediNext.