European Options in BS Markets - Computational Methods for Quantitative Finance - page 59

Define the weighted Sobolev space

(G) · ρ , C 0 W ρ = (4.20)

where the weighted Sobolev norm · ρ is defined by

R

2

(s 2 ρ

2 2 ) d s,

ρ :=

| | +| | ≤ ≤

ϕ ∂ s ϕ ϕ 0 ρ 1 . (4.21)

0

C 0

For ϕ,φ ∈ (G) , we define the bilinear form

2 σ 2

s 2 ρ ∂ s ϕ∂ s φ d s + ρσ 2

R R

1

a CEV
ρ

s 2 ρ − 1 ∂ s ϕφ d s

(ϕ, φ) :=

0 0

r r

R R

− s∂ s ϕφ d s + ϕφ d s. (4.22)

0 0

The variational formulation of ( 4.19 ) is based on the triple of spaces V = W ρ →

L 2 (G) = H ∗ W ρ = V ∗ , and reads: H = →

L 2 (J H 1 (J L 2 (G)) such that Find v ∈ ; W ρ ) ∩ ;

a CEV

+ = ∀ ∈

(∂ t v,w) (v, w) 0 , w W ρ , a.e. in J, (4.23)

ρ

v( 0 ) = g.

To establish well-posedness of ( 4.23 ), we show continuity and coercivity of the
bilinear form a CEV
ρ

( · , · ) on W ρ .

Proposition 4.5.1 Assume r> 0. There exist C 1 ,C 2 > 0 such that for ϕ,φ ∈ W ρ

a CEV
ρ

| (ϕ, φ) |≤ C 1 ϕ ρ φ ρ ,ρ ∈[ 0 , 1 ]\{ 1 / 2 } , (4.24)

1
2 .

a CEV

2

(ϕ, ϕ) ≥ C 2 ϕ

ρ ,

0 ≤ ρ ≤ (4.25)

ρ

C 0

Proof Let ϕ ∈ (G) . By Hardy's inequality, for ε = 1, ε> 0, and any R> 0

2 d s 2 d s

R

1
2

R

1
2

2

s ε − 2 s ε | ϕ | ≤ | ∂ s ϕ | , (4.26)

| − |

ε 1

0 0

we find with ε = 2 ρ = 1, that

2

s 2 ρ − 1 ∂ s ϕφ d s ≤ s 2 ρ (∂ s ϕ) 2 d s s 2 ρ − 2 φ 2 d s

R R

1

R

1
2

0 0 0

2

≤ ϕ ρ

|

φ ρ

| 2 ρ − 1