Parabolic Variational Inequalities - Computational Methods for Quantitative Finance - page 277

→

Note that, as k 0,

V ∗ strongly . G 0 → u 0 ∈ H ,H 0 → 0in (B.19)

Since 0 ∈ K , we may choose in ( B.17 ) v = 0 and get

2 2 2 2

∗

u 1

H +

k u 1

V ≤

C( G 0

H +

H 0 ), (B.20)

V ∗ . We claim that
u 1 − P u 0

where · ∗ denotes the norm in

2 2

H + k u 1 V →

0as k → 0 . (B.21)

To prove ( B.21 ), we note that u 1 ∈ K . Also, by the definition of P in Remark B.1.1 ,

we have

(u 0 − P u 0 ,u 1 − P u 0 ) ≤ 0 ,

hence

2

u 1 − P u 0

H =

(u 1 − P u 0 ,u 1 − P u 0 ) ≤ (u 1 − u 0 ,u 1 − P u 0 ). (B.22)

By ( B.4b ) with λ = 0, ( B.9c ) and ( B.9e ), ( B.17 ), ( B.22 ), it follows for every v ∈ K

2 2

u 1 − P u 0

H +

u 1 −

V ≤

(u 1 − u 0 ,u 1 − P αk v u 0 )

+ A

(u 1 − v),u 1 −

V ∗ , V

k v

≤ (u 1 − u 0 )

+ k u 1 − u 0 + k A u 1 ,u 1 − v V ∗ , V
− k A v,u 1 − v V ∗ , V

u 0 ,v − P

( B.17 )
≤ u 1 − u 0 H v − P u 0 H

+ (G 0 − u 0 ,u 1 − v)

√ k √ k

+ H 0 − A v,u 1 − v V ∗ , V .

Therefore, it holds for all v ∈ K that

2 2

V

u 1 − P u 0

H +

k u 1 − u 0

C

≤

u 1 − u 0 H

− P

u 0 H

v

. (G 0 − √ k H 0 − √ k

v

+ u 0 ,u 1 − v) + A v,u 1 − (B.23)

V ∗ , V

∈ K · H , there is a sequence Since P u 0 { v k } k> 0 ⊂ K such that, as k → 0,

2

k v k and obtain, after

passing to the limit and using ( B.19 ), ( B.20 ), the assertion ( B.21 ). We have

v k

V →

0, and v k − P u 0 H → 0. We choose in ( B.23 ) v =

Lemma B.3.1 For any fixed n> 0, there exists C n > 0 such that , as k → 0,

C n

S( 0 ,nk) .

2 2 2 2

u k,n − P u 0

H +

k u k,n

V ≤

u 0

H +

f (B.24)

Moreover , as k → 0,

2 2

u k,n − P u 0

H +

k u k,n

V →

0 (non-uniformly in n) . (B.25)