Parabolic Variational Inequalities - Computational Methods for Quantitative Finance - page 282

{ } k> 0 by

Proposition B.3.10 Define the step-functions u k (t)

u k (t) J k,m := u k,m ,m =

0 ,...,M − 1 , (B.55)

and analogously f k (t) . Assume 0 0 ,α> 0. Then ,

u k (t + k) I( 0 ,T ) ≤ C(α) w 0 H + f k (t) S( 0 ,T + k) ,

∈ K and ( B.4b ) with λ =

(B.56)

√ k

C(α) S( 0 ,T + k) .

(B.57)

u k (t + k) I( 0 ,T ) ≤ u k, 1 H + u k, 1 V + f k (t)

Proof We choose 0 = v ∈ K in ( B.12b ). Since, for m> 0, u k,m ∈ K and we have

( B.29 ), ( B.30 ) with w m = u k,m , f k,m = p m + q m ,( B.56 ) and ( B.57 ) follow then

from ( B.53 ), ( B.54 ).

We also have a bound for U k (t) in ( B.13a ), ( B.13b ).

Lemma B.3.11 Assume ( B.4b ) with λ . The semi-discretization

( B.11 ), ( B.12a ), ( B.12b ) of PVI ( B.9a )-( B.9e ) is stable in the sense that for T

= 0 and 0 ∈ K

= Mk

C S( 0 ,T + 2 k) . U k (t) I( 0 ,T ) ≤ u 0 H + f(t) (B.58)

Proof For T = Mk ,wehave

f k (t) S( 0 ,T ) ≤ f(t) S( 0 ,T + k) ; U k (t) I( 0 ,T ) ≤ u k (t) I( 0 ,T + k) .

Inserting this into ( B.56 ), we get ( B.58 ).

Remark B.3.12 Note that we could also obtain an a priori bound for U k (t) from
( B.57 ): here ( B.25 ) with n

= 1 would imply

C P S( 0 ,T + 2 k) .

U k (t) I( 0 ,T ) ≤ u 0 H + f(t) (B.59)

We are now able to show our first main result. To state it, we define, for f k (t) as
in ( B.55 ),

f k (t) − f(t) S( 0 ,T + 2 k) + f(t + k) − f(t) S( 0 ,T + 2 k)

√ k

E(k,T ) :=

+

u k, 2 − u k, 1 H + u k, 2 − u k, 1 V + u k, 1 − P u k, 0 H .

(B.60)

Lemma B.3.13 We h ave f o r T> 0, as k 0,

kU k (t) I( 0 ,T ) ≤ CE(k,T).

= T/M →

(B.61)

Proof We note that due to its definition ( B.13a ), ( B.13b ), kU k (t) | J k,m = u k,m + 2 −

= = +

u k,m + 1 . Selecting v u k,m + 2 in ( B.12b ) and also v u k,m + 1 in ( B.12b ) with m 1

in place of m gives the two inequalities:

u k,m + 1 − u k,m + k A u k,m + 1 − f k,m ,u k,m + 1 − u k,m + 2 V ∗ , V ≤ 0 ,

− u k,m + 2 − u k,m + 1 + k A u k,m + 2 − f k,m + 1 ,u k,m + 1 − u k,m + 2 V ∗ , V ≤ 0 .