Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 221

d ) . Then , there is C 1 > 0
E (g(x + Z 1 ,T )) − E (g(x + Y 1 ,T )) ≤ C 1

C 2 (

∈ R

Proposition 14.7.4 Assume g

ε

d

z j

2 ν j ( d z j ),

d .

∀ x ∈ R

− ε

j = 1

(14.37)

d ) and
R

C 4 (

Furthermore , assume g ∈ R | z | ν( d z) < ∞ . Then , for some C 2 > 0

d

ε

d

E

Y 2 ,T )) ≤

z j

3 ν j ( d z j ),

Z 2 ,T ))

d .

(g(x + − E (g(x + C 2 ∀ x ∈ R

− ε

j = 1

(14.38)

Proof Consider the Taylor series expansion of g(x) at x 0

1
2 (x − x 0 ) · D 2 g(x 0 )(x − x 0 ) + O ( | x − x 0 |

3 ),

g(x) = g(x 0 ) +∇ g(x 0 ) · (x − x 0 ) +

where D 2 g is the Hessian matrix of g . Define R ε Y 1 . The Lévy process R ε

has Lévy measure ν ε and is independent of Y 1 . Since Z ε,j

Z 1 −

=

and Y ε,j

1 ,T , j

= 1 ,...,d ,

1 ,T

E (R ε,j
T

have the same expected value, we have ) = 0, j = 1 ,...,d . Thus, we obtain

E (g(x + Z 1 ,T )) − E (g(x + Y 1 ,T ))

d

∂ j g x

Y 1 ,T

d d

R ε, T

E R ε, T + E R ε,j

T

=

1 E

+

1 O j = j = 1 k =

R ε,j
T

2 .

d

≤ C 1

1 E

j =

Equation ( 14.37 ) follows from

ε

R ε,j
T

2

z j

2 ν j ( d z j ),

E = j = 1 ,...,d.

− ε

Furthermore, Y 2 ,t = Y 1 ,t + (γ 2 − γ 1 )t , where the standard Brownian motion

Σ ε W t +

W is independent of Y 1 .Weset x = x + (γ 2 − γ 1 )T and obtain

E

Y 2 ,T ))

Z 2 ,T )) (g(x + − E (g(x +

E

Y 2 ,T )) Z 1 ,T )) Y 1 ,T )) Y 1 ,T )) = (g( x + − E (g( x + + E (g( x + − E (g(x +

∂ j ∂ k g

1 ,T

R ε,j
T

3

d d d

E |

1
2 E

Y ε,j R ε,j

T

R ε,k
T

= x + E +

1 O

|

j = 1 k = 1 j =

∂ j ∂ k g

1 ,T

4

d d

E |

1
2 E

Y ε,j

− x + Q ε,jk + O Σ ε W T |

j = k =

1 1

ε

2 ν j ( d z j ) 2 .

ε

d

z j z j

3 ν j ( d z j )

≤ C 2 +

− ε − ε

j = 1