sparse grid method in the libor market model. option valuation and the

T
0 , 0
T
0 , 1
T
0 , 2
T
0 , 3
|i|
1 = 0
T
1 ,0
T
1 , 1
T
1 ,2
· · ·
|i|
1
= 1
T
1 ,3
T
2 ,0
T
2 , 1
·
·
T
2 ,2
T
2 , 3
|i|
1 = 2
T
3 ,0
|i|
1 = 3
|i|
1 = 4
T
3 , 1
|i|
1
= 5
T
3 ,2
|i|
1
= 6
T
3 , 3
Figure 5.2: Hierarchical representation.
spaces with a smaller |l| 1 value. For example, applying (5.4) to S 3 should yield linear
images of spaces labeled T 3,0 or, T 0,3 on Figure (5.2). Note that, unlike S 3 , the support
of the basis functions of T 3 is not overlapping.
Now we can use (5.4) to obtain the final n-level hierarchy of subspaces S ∗ n:
S ∗ n =
n⊕
· · ·
l 1=0
n⊕
T l =
l d =0
⊕
|l| ∞ ≤n
T l (5.5)
’≤’ corresponds to index-wise relation in the subscript of the direct sum operator. Figure
(5.2) illustrates such multi-level subspace decompostion S3 ∗ in two dimensions.
Figure (5.3) shows how hierarchical subspaces T l1,l 2
reconstruct Ω 3,3 , where each tuple
relates to the subspace index. As stated in the beginning of this chapter, our original
intent is to approximate a PDE solution space. The equation of our interest, given
by (3.36), is subject to Dirichlet boundary condition, which implies no degrees of freedom
exist on the boundaries of solution domain. For this specific reason, subspaces
with l = 0 index components have not been included in the hiearchical representation
of (5.3).
39