Parabolic Variational Inequalities - Computational Methods for Quantitative Finance - page 279

αk

M 1

2

2
V

X M :=

≤ m ≤ M { w m H } ,

max M :=

1 w m

(B.31a)

1

m =

M − 1 M − 1

2

k
α

2 2

∗

P M := k

0

p m H , := w 0

H +

q m .

M

m = m = 0

(B.31b)

Then it holds

Q 2 M ) 2 .

(P M + max (X M ,Y M ) ≤ P M + (B.32)

Proof By ( B.30 ) and ( B.4b ) with λ = 0,

2
V

2 (w m + 1 − w m ,w m + 1 ) + 2 kα w m + 1

≤ 2 k p m H w m + 1 H + 2 k q m ∗ w m + 1 H

k
α

2 2

H

≤ 2 k p m H w m + 1 H + q m

∗ +

αk w m + 1 .

Using here

2 2

H

(w m + 1 − w m ,w m + 1 ) ≥ w m + 1

H −

w m (B.33)

and summing from m = 0 ,...,M − 1, we get

M − 1

2

Y M ≤ Q 2 M ≤ Q 2 M . w M

H +

2 k

0

w m + 1 H p m H + 2 X M P M + (B.34)

m =

For M ≥ 1, we infer from ( B.34 ) the bound

Y M ≤

Q 2 M , 2 X M P M + (B.35)

≤ ≤

and also, for 1 m M ,

2

Q 2 M . w m

H ≤

2 X M P M + (B.36)

Hence, from ( B.36 ),

X 2 M =

2

Q 2 M ,

M

w m

H ≤

2 X M P M +

max

1 ≤ m ≤

and, combining with ( B.35 ), we get ( B.32 ).

Remark B.3.5 We can also bound the jumps w m + 1 − w m :

M − 1

Q 2 M ) 2 .

2

(P M +

0

w m + 1 − w m

H ≤

P M +

(B.37)

m =

To obtain this, replace ( B.33 )by

2 2 2

H

2 (v − w,v) = v − w

H +

v

H −

w .

We also need a continuous analogue of Lemma B.3.4 .