## 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 4d20 : 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(h^{2}) . The piecewise-d-linear sparse grid approximation requires only
degrees of freedom,
while the error is .

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 ItÃ´'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.