Multidimensional Feller Processes - Computational Methods for Quantitative Finance - page 257

Note that the (non-constant) symbol of the infinitesimal generator of X does
generally not coincide with the characteristic exponent of the process X , due to the
spatial inhomogeneity of X . We illustrate the preceding, abstract developments with
an example related to the so-called tempered-stable class of Lévy processes which
were advocated in recent years in the context of financial modeling.

Example 16.4.5 (Feller-CGMY) We consider a d -dimensional Feller process with
Clayton Lévy copula

2 2 − d

ϑ −

1
ϑ

d

ρ 1 ,

F(u 1 ,...,u d ) =

1 | u i |

} − ( 1

− ρ) 1

{ u 1 ,...,u d ≥ 0 { u 1 ,...,u d ≤ 0 }

i =

where ϑ> 0, ρ ∈[ 0 , 1 ] together with CGMY-type densities

C(x) e − β i (x) | z |
|

,

e − β i (x) | z |
|

=

Y i (x) 1 { z< 0 } +

k i (x, z)

Y i (x) 1 { z> 0 } 1 + 1 +

| |

z z

with smooth an d bounded functions C(x) > 0, β i (x) > 0, β i

(x) > 1 and 0 <

Y i <Y i (x) 1 ,...,d . We assume the

Gaussian component Q(x) to be positive semidefinite, smooth and bounded. The
drift γ(x) is assumed to be smooth and bounded. It is easy to see that this market
model satisfies properties (i), (ii), (iv)-(vi) of the above definition. Properties (iii)
and (vii) follow analogously to the proof of [163, Proposition 2.3.7].

≤ Y i < 2, Y i (x) sufficiently smooth, for i =

16.5 Variational Formulation

We first prove a sector condition for admissible market models, which is crucial for
the proof of well-posedness of the pricing equation. Subsequently, well-posedness
results for certain types of pricing equations are given. These equations are of
parabolic type, with the highest order operator being the diffusion or jump part of
the generator of the market model.

16.5.1 Sector Condition

The sector condition for the symbol ψ(x,ξ) of a Feller process X is one of the main
ingredients for proving well-posedness of the initial boundary value problems for
the PIDEs arising in option pricing problems. The sector condition reads:

d 2 m (x) .

∃ C> 0 . ∀ x,ξ ∈ R : ψ(x,ξ) + 1 ≥ C ξ (16.21)

Theorem 16.5.1 Let X be an admissible time-homogeneous market model process
taking values in

d

R with characteristic triplet (b(x), Q(x), k(x, z) d z) . Then , there