## Preliminary list of abstracts

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*Keywords:*Sparse grids; Preconditioner; Generating system; Finance

- Hullmann, Alexander (Institute for Numerical Simulation, University of Bonn, Germany)

The deterministic numerical valuation of basket options on underlyings with jumps results in a multi-dimensional partial integro-differential equation. In order to efficiently discretise the equation in variational form, we use a sparse grid generating system based on linear splines with only $O(N (\log N)^{d-1})$ degrees of freedom, as opposed to $O(N^d)$ in the full-grid case. Here, $N=h^{-1}$ is the finest resolution in one direction and $d$ the space dimension.

To obtain a fast and efficient overall solution method, several
numerical issues need to be addressed. In contrast to the local part of
the PIDE-operator, the integral-operator leads to a dense matrix after
discretisation. We will generalise a finite difference recurrence
formula for the Kou jump-diffusion model to the multi-dimensional
Galerkin case, which enables us to evaluate the integral in
* linear* time.

Secondly, also the local part of the PIDE-operator will not straightforwardly lead to a sparse operator matrix due to the non-local nature of sparse grid basis functions. But applying the unidirectional principle to sparse grid generating systems, we can evaluate the operator in linear time, using only restrictions, prolongations and non-hierarchical finite-element operators.

The corresponding systems of linear equations must then be solved in a fast way by, e.g. additive multilevel methods. As generating system-based preconditioners for sparse grids exhibit a suboptimal condition number of the order $O((\log N)^{d-1})$ only, we discuss new optimal preconditioners that result in $O(1)$ condition numbers. Here, multiresolution norm equivalences are used without specifically discretising the detail spaces, while the practical advantages of the generating system approach are preserved. Altogether, we thus obtain a solution method for the sparse grid discretisation of the PIDE in optimal complexity.