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  or on sparse tensor product approximation spaces  ,  .
Sparse grids were introduced by Zenger  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 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 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.