Stochastic Volatility Models - Computational Methods for Quantitative Finance - page 111

1
8 (y∂ x ϕ,y∂ x φ) +

1
2 β 2 (∂ y ϕ,∂ y φ) +

1
2 ρβ(y∂ x ϕ,∂ y φ)

a κ (ϕ, φ) =

1
2 [ ρβ −

1
8 [

+ 2 r ] (∂ x ϕ,φ) + 1 − 4 βκρ ] (y∂ x ϕ,yφ)

1
2 [

1
2 [

2 β 2 κ β 2 (y − 1 ∂ y ϕ,φ) − − α ] (y∂ y ϕ,φ) − 4 αm − ]

1
2 κ

β 2 κ − [ − α ] (yϕ, yφ) −[ 2 καm − r ] (ϕ, φ)

9

=: b k (ϕ, φ). (9.22)

k = 1

Define the weighted Sobolev space

(G) · V , C 0 V := (9.23)

where the closure is taken with respect to the norm

1

2 2 2 2

y 2 v

v

V :=

y∂ x v

L 2 (G) +

∂ y v

L 2 (G) +

+

L 2 (G) .

(9.24)

Theorem 9.3.1 Assume that 0 <κ<α/β 2 and that

4 αm/β 2 >ρ 2 .

− | − |

1 2 1

Then , there exist constants C i > 0, i = 1 , 2 , 3, such that for all ϕ,φ ∈ V one has

a κ (ϕ, φ) | |≤ C 1 ϕ V φ V ,

a κ (ϕ, ϕ)

2 2

≥ C 2 ϕ

V −

C 3 ϕ

L 2 (G) .

C 0 (G) . We first show continuity of the bilinear form a κ (

Proof Let ϕ,φ ) .

The only terms in ( 9.22 ) which cannot be estimated directly by an application of
Cauchy-Schwarz are (∂ x ϕ,φ) and (y − 1 ∂ y ϕ,φ) . For both terms, the continuity fol-
lows from the Hardy inequality (4.26)
y − 1 φ L 2 (G) ≤

∈ · , ·

2 ∂ y φ L 2 (G) . (9.25)

Indeed,

2 [

] y − 1 ∂ y ϕ,φ

1

β 2 4 αm −

( 9.25 )
≤ ∂ y ϕ L 2 (G) ∂ y φ L 2 (G) , ≤ C ∂ y ϕ L 2 (G) y − 1 φ L 2 (G)

and similarly
1

2 [

] ∂ x ϕ,φ ≤ y − 1 φ ρβ − 2 r C y∂ x ϕ L 2 (G) L 2 (G) ≤ C y∂ x ϕ L 2 (G) ∂ y φ L 2 (G) .

a κ (ϕ, φ)

To this end, it follows from the triangle inequality V .To

prove the Gårding inequality, we consider each term in ( 9.22 ) separately. We have,
since (y∂ y ϕ,ϕ) =

| |≤ C 1 ϕ V φ

1

2 (∂ y (ϕ 2 ), y) =−

1 2

2 ϕ

L 2 (G) ,