Parabolic Variational Inequalities - Computational Methods for Quantitative Finance - page 284

∈

This gives ( B.62b ), if we take the infimum over all decompositions of f S( 0 ,T)

of the form f h as in ( B.15a ), provided u 0 satisfies also ( B.15b ).

To show ( B.62a ) for general f

= g +

,weuse( B.21 ), ( B.25 ) to bound
the second row of ( B.60 ), and approximate a general f

∈ S(a,b),u 0 ∈ H

∈ ; H +

S( 0 ,T) from BV (J )

H 1 (J ; V ∗ ) to bound the first row of ( B.60 )by o( 1 ) as k

→ 0.

We are now ready to prove the existence Theorem B.2.2 . To this end, we show
that the family

{ } k> 0 is Cauchy in I( 0 ,T) . More precisely, there is C> 0 such

U k (t)

that

C E(k,T )

E(h,T ) .

U k (t) − U h (t) I( 0 ,T ) ≤ + (B.63)

{

This and ( B.62a ), ( B.62b ) imply that U k } k> 0 is Cauchy in I( 0 ,T) and that there is

U = lim k → 0 U k (t) ∈ I( 0 ,T) with

CE(k,T ). (B.64)
We note that ( B.64 ) with ( B.62a ), ( B.62b ) gives an err or estimate for ( B.12a ),
( B.12b ). To prove ( B.63 ), we recall the definition ( B.55 )of u k (t) and we also define

U(t) − U k (t) I( 0 ,T ) ≤

u k (t) = k (t) u k (t) + ( 1 − k (t)) u(t + k), (B.65)

where

k (t) := m + 1 − t/k ∈[ 0 , 1 ] ,t ∈ J k,m =[ mk, (m + 1 )k ] . (B.66)

Then it follows from ( B.12a ), ( B.12b )thatforall t ∈ J holds

v u k + . (B.67)

To prove ( B.63 ), we proceed as follows: we rewrite ( B.67 ) in terms of U k (t) in
( B.13a ), ( B.13b ) with small right hand side. Then ( B.63 ) will be obtained by appli-
cation of Lemma B.3.6 to U k (t)

Au k (t + k) − f k (t), u k (t + k) −

V ∗ , V ≤

0 ∀ v ∈ K

− U h (t) .

H
u k (t) − u k (t + k) H ≤ u k (t) − u k (t + k) H = ku k (t) H ,

≤

k ≤

We have from 0 1 that in

(B.68)

and also in V , and for every (a, b) ⊆ J , that

u k (t) I(a,b) ≤ u k (t) I(a,b) + u k (t + k) I(a,b) ≤ 2 u k (t) I(a,b + k) . (B.69)

Lemma B.3.15 Fo r a n y v ∈ K , the following holds :

U k (t) + AU k (t) − f k (t + k), U k (t) − v V ∗ , V
≤ β kU k (t) V U k (t) − v V
− k (t) U k (t) k),kU k (t)

+ A u k (t + k) − f k (t + . (B.70)

V ∗ , V

Proof We have by ( B.4a )
A − V ∗ , V ≤ A + − V ∗ , V

U k (t), U k (t) v u k (t k),U k (t) v

+ A u k (t + k) − A u k (t + k),U k (t) − v V ∗ , V

≤ A V ∗ , V

+ β u k (t + k) − u k (t + k) V U k (t) − v V

u k (t + k),U k (t) − v

=: I + II .