Parabolic Variational Inequalities - Computational Methods for Quantitative Finance - page 286

Lemma B.3.18 Fo r u 0 ∈ K · H and f

∈ S( 0 ,T) , and any h, k > 0, one has

C E(h,T )

E(k,T ) ,

U h (t) − U k (t) I( 0 ,T ) ≤ + (B.74)

with E(h,T ) as in Lemma B.3.14 . In particular , { U h (t) } h> 0 is Cauchy in I( 0 ,T)

and there exists

= ∈

u lim

h → 0

U h (t) I( 0 ,T).

Proof We choose in ( B.73 ) v U h (t) for some h> 0, and then exchange in the

resulting inequality the roles of k and h . Adding the resulting two inequalities for
the difference w(t)

=

U h (t) , we ge t a n inequality of the type considered in
Lemma B.3.6 , with s(t) replaced by s(t)

:= U k (t) −

f(t) : To determine r(t) in
( B.40 ), we estimate the last term in the bound ( B.73 )asfollows:using0

+ f k (t + k) −

≤ k (t) ≤ 1

and

U k (t), kU k (t)

k (t) − V ∗ , V ≤ 0 , (B.75)

we have

A

V ∗ , V

kU k (t) 0 ≤ r(t) := u k (t + k) − f k (t + k), −

≤ β H kU k (t) u k (t + k) V + f k (t + k) V .

Hence, in ( B.42 ),

T

R(T ) = r(t) d t

0

Const S( 0 ,T + k) · kU k (t)

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

From ( B.56 ), we get
R(T )

Const S( 0 ,T + k)

kU k (t) ≤ w 0 H + f k (t) I( 0 ,T ) ,

and ( B.61 )gives

Const S( 0 ,T + k) E(k,T ).

R(T ) ≤ w 0 H + f k (t) (B.76)

To estimate the value of w( 0 ) H in ( B.76 ) and in ( B.43 ), we use that, by

Lemma B.3.2 ,

w( 0 ) H = (u k, 1 − P u 0 ) − (u h, 1 − P u 0 ) H ≤ E(k,T ) + E(h,T ),

which, inserted into Lemma B.3.6 , implies the assertion.

We can now give the proof of Theorem B.2.2 (i). Lemma B.3.18 established that

u h } h> 0 is Cauchy in I( 0 ,T) , hence in particular in L 2 (J ) and in L ∞ (J { ; V ; H ) .

C 0 (J

Th erefore, u(t) ∈ ; H ) and u( 0 ) = lim k → 0 U k ( 0 ) = lim k → 0 u k, 1 = P u 0 ∈

· H , which is the third line in ( B.6 ).
To show that u(t) is a solution of the PVI, pick in ( B.73 ) v(t) satisfying ( B.7b ),
and pass in ( B.73 ) to the limit k →

K

0, implying the second line of ( B.6 ); since K is

u in L ∞ (J

closed in ) ,wealsohavethefirstlineof( B.6 ).

The uniqueness and Theorem B.2 (ii) will follow from

H and U h → ; H