Sensitivities and Greeks - Computational Methods for Quantitative Finance - page 148

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(η 0 ) with respect to η 0 ∈ S η We also introduce the derivative of

A (δη)ϕ := D η 0 A (η 0 )(δη)ϕ

s A

(η 0 )ϕ ,ϕ

1

:= lim

s → 0 +

(η 0 + sδη)ϕ − A ∈ V ,δη ∈ C .

A (δη) ∈ L ( V , V ∗ ) with V

We assume that a real and separable Hilbert space satis-

fying

V ⊆ V ⊂ H = H ∗ ⊂ V ∗ ⊆ V ∗ .

V ⊆ V

We further assume th at there exists a real and separable Hilbert space such

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∈ V ∗ , ∀ ∈ V

that v v . We have the following relation between D η 0 u(η 0 )(δη) and u .

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( V , V ∗ ) , Lemma 11.2.1 Let (δη) ∈ L ∀ δη ∈ C and u(η 0 ) : J → V , η 0 ∈ S η be the

unique solution to

d , 0 )( 0 , d .

− A = × R · = R

∂ t u(η 0 ) (η 0 )u(η 0 ) 0 in J ) g(x) in (11.6)

Then , u(δη) solves

= A

d , d .

(11.7)

∂ t u(δη) − A (η 0 ) u(δη) (δη)u(η 0 ) in J × R u(δη)( 0 , · ) = 0 in R

Proof Since u(η 0 )( 0 ) = g does not depend on η 0 , its derivative with respect to η

is 0. Now let η s := η 0 + sδη , s> 0, δη ∈ C . Subtract from the equation ∂ t u(η s )(t) −

A (η s )u(η s )(t) = 0Eq.( 11.6 ) and divide by s to obtain

s u(η s )(t)

u(η 0 )(t) −

s A

(η 0 ) u(η s )(t) ∂ t 1

1

− (η s ) − A

(η 0 ) u(η s )(t) u(η 0 )(t) =

1
s A

− −

0 .

Taking lim s → 0 + gives Eq. ( 11.7 ).

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: V × V → R

We associate to the operator (δη) the Dirichlet form a(δη ;· , · )

which is given by

a(δη ; u, v) =− A (δη)u, v V ∗ , V .

The variational formulation to ( 11.7 ) reads:

L 2 (J H 1 (J Find u(δη) ∈ ; V ) ∩ ; H ) such that

a η 0 ; u(δη), v =− a δη u(η 0 ), v ,

(∂ t u(δη), v)

H +

; ∀ v ∈ V , (11.8)

u(δη)( 0 ) = 0 .

Note that ( 11.8 ) has a un ique solution u(δη) ∈ V due to the assumptions on

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a(η 0 ;· , · ) , and u(η 0 ) ∈ V .