AMFR-W Numerical Methods for Solving High-Dimensional SABR/LIBOR PDE Models

Boundary corrections on multi-dimensional PDEs

“…Following the ideas given in [13], a space discretization is performed on a uniform spatial grid on Ω. Given N integers (M 1 , .…”

“…The handling of these differences becomes very cumbersome when the dimension is greater than 3, especially when we have to deal with the values on the boundary. In [13,Lemma 3.1], a map is defined that assigns an integer J to each multi-index (j 1 , . .…”

“…Here we suppose that time-dependent Dirichlet conditions are imposed in all the boundary of Ω. Then, we apply [13,Lemma 3.1] to define the two following maps, with…”

“…Although it is not a proper semi-discretized IVP, since we need to involve the invariant (9), from the theoretical point of view this is only a simpler way to write the usual semi-discretized IVP with these differences (see, for instance, [11]) when PDEs of high dimension are solved. This is the option used in [13] to integrate satisfactorily a model with spatial dimension six.…”

“…In particular, we are interested in AMF-type W-methods (based on Approximate Matrix Factorization) which avoid the solution of non-linear equations and only require the solution of linear systems with tridiagonal matrices, as it can be seen in [7], [6], [5]. They have also been successful in applications to the case of variable coefficients in [9] and multi-dimensional problems (N > 3) in [13].…”