Team Mathfi

Overall Objectives
Scientific Foundations
Application Domains
New Results
Contracts and Grants with Industry

Section: New Results

Keywords : sparse grids, finite element, adaptive finite elements, lattice-based methods.

Sparse grids methods for PDEs in Mathematical Finance

Participants : Y. Achdou, D. Pommier.

Recent developments have shown that it may be possible to use deterministic Galerkin methods or grid based methods for elliptic or parabolic problems in dimension d, for 4$ \le$d$ \le$20 : these methods are based either on sparse grids [65] or on sparse tensor product approximation spaces [66] , [85] .

Sparse grids were introduced by Zenger [87] in order to reduce the number of degrees of freedom of discrete methods for partial differential equations. Standard piecewise linear approximations need O(h-d) degrees of freedom, (h is the mesh step), and produce errors of the order of O(h2) . The piecewise-d-linear sparse grid approximation requires only Im17 ${O\mfenced o=( c=) h^{-1}{|logh|}^{d-1}}$ degrees of freedom, while the error is Im18 ${O\mfenced o=( c=) h^2{|logh|}^{d-1}}$ .

D. Pommier and Y. Achdou study these methods for the numerical solution of diffusion or advection-diffusion problems introduced in option pricing. They consider the case of a European vanilla contract in multifactor stochastic volatility models. Using Itô's formula, they obtain a 4 dimensional PDE problem, which they solve by means of finite differences on a sparse grid. They compare this accuracy and computing time to those of standard Monte Carlo methods.

A Cifre agreement on this subject between Inria and BNP-Paribas is engaged on this subject for the PhD thesis of David Pommier.


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