For the solving a $d$-dimensional SDE with a sparse grid algorithm related to an option pricing problem I ended up having to solve the following integral equation.
$g_{\bar{l},\bar{i}} = \int_D p(e^{x_1},\dots, e^{x_d}) \varphi_{l_1,i_1}(x_1) \dots \varphi_{l_d,i_d}(x_d) d(x_1,\dots,x_d)$
Whereas
- $\bar{l} = (l_1, \dots, l_d)$ stands for the level of current refinement and $\bar{i} = (i_1, \dots, i_d)$ is a multi-index referring to the index of the grid point in the sparse grid. This yields that the tupel $(\bar{l},\bar{i})$ refers to grid point in the sparse grid.
The sparse grid has no boundary points and is constructed via the hierarchial basis approach. This means I already calculated the given grid points of the sparse grid. The are also the notes of my quadrature. My problem is that I do not know on how to choose proper weights for my quadrature.
So basically my questions come down to: Assuming the grid points of a sparse grids are given through a hierarchical basis approach, how am I suppose to choose the weights of a quadrature if we assume that the nodes of the quadrature are the given sparse grid points.
Further information, but not relevant for solving the problem.
- function $p(..)$ is a given payoff
- the $\varphi_{l_k,i_k}$ refer the standard construction of sparse grids through the hierarchical basis. Namely speaking for the general hat function defined through $\varphi(x) = max\{0,1-|x|\}$ we have that $\varphi_{l_k,i_k}(x_k) = \varphi(2^{l_k}x_k-i_k))$