Multidimensional Feller Processes - Computational Methods for Quantitative Finance - page 253

H

Theorem 16.3.7 Let (T t ) t ≥ 0 be a Feller semigroup on a Banach space with gen-

erator be a Bernstein function and let (η t ) t ≥ 0 denote the

associated convolution semigroup on

A , let f : ( 0 , ∞ ) → R

0 , ∞ ) . Define T t u for u ∈ H

R supported on [

by the Bochner integral

∞

T t u

= T s uη t ( d s). (16.17)

0

Then , the integral is well-defined and (T t

) t ≥ 0 is a Feller semigroup on H .

Proof See [94, Theorem 4.3.1 and Corollary 4.3.4].

The representation ( 16.17 )of T t u can be used for numerical methods, but it in-
volves the approximation of an integral over a possibly semi-infinite interval, which
can be very costly if the integrand is not well-behaved. The generator

f of the
semigroup (T t ) t ≥ 0 is a PDO and, in the case of a spatially homogeneous semi-
group (T t ) t ≥ 0 , its symbol corresponds to a Lévy process X given as follows.

A

Theorem 16.3.8 Let (T t ) t ≥ 0 be a Feller semigroup with generator with constant

symbol ψ(ξ) and f(x) as in the previous theorem , then the symbol ψ f (ξ ) of the
generator

A

of the semigroup (T t ) t ≥ 0 is given as
ψ f (ξ )

f

A

= f(ψ(ξ)).

Proof See [3, Proposition 1.3.27].

The same characterization does not hold in the case of a more general subordi-
nated process. We obtain the following representation for

f .

A

Theorem 16.3.9 Let f and (T t ) t ≥ 0 be as in Theorem 16.3.7 . Fo r a l l u ∈ D( A ) , we

f ) and

∈ A

have u D(

∞

f u

A = + A +

R λ A

au b u uμ( d λ),

0

, i . e . R λ u = (λ − A ) − 1 u and a , b and μ(ds) where R λ denotes the resolvent of A

are defined in Theorem 16.3.5 .

Proof See [96, Theorem 2.15].

This resolvent representation is of limited use for numerical computation, and
we therefore aim at a characterization of the symbol of the PDO

f . We consider a
certain symbol class as in [96]. Let L(x, D) be the differential operator given by

A

d

∂ 2
∂x i ∂x j +

L(x, D) =− a i,j (x) c(x),

i,j = 1