Multidimensional Feller Processes - Computational Methods for Quantitative Finance - page 258

d

∈ R ∞

exists a constant C> 0 such that for all x and ξ sufficiently large

j = 1 |

d

Y j (x j ) ,

ψ(x,ξ) ≥ C ξ | (16.22)

=

2 in the case Q 0 ≥

where Y j (x j ) Q> 0.

Proof The proof mainly follows the arguments of [163, Proposition 2.4.3]. We refer
to [135, Theorem 5.1.6].

16.5.2 Well-Posedness

For an admissible time-homogeneous market model X , we can derive a PDO and
PIDE representation and prove well-posedness of the weak formulation of the prob-
lem on a bounded domain. Due to no arbitrage considerations, we require the con-
sidered processes to be martingales under a pricing measure Q . This requirement

can be expressed in terms of the characteristic triplet.

Lemma 16.5.2 Let X be an admissible time-homogeneous market model with char-
acteristic triplet (b(x), Q(x), ν(x, d z)) and semigroup (T t ) t ≥ 0 further let T t (e x j )<
∞ 1 ,...,d . Then , e X j is a

hold for t ≥ 0, j = Q -martingale with respect to the

canonical filtration of X if and only if
Q jj (x) 2
2

(e z j

+ + − − = ∀ ∈ R

b j (x) 1 z j )ν j (x, d z j ) 0 x . (16.23)

0 = y ∈R

Proof This is a direct consequence of [61, Sect. 3].

Remark 16.5.3 Note that without the assumption of finiteness of exponential mo-
ments of the processes X j , the processes e X j , j 1 ,...,d would generally only

be local martingales. For Lévy processes, exponential decay of the jump measure
implies the existence of exponential moments, cf. [143, Theorem 25.3]. This is not
obvious for general Feller processes. Recently, Knopova and Schilling have proved
in [106] the finiteness of exponential moments for a certain class of Feller processes
assuming exponential decay of the density of the jump measure.

=

We are now able to derive a PDO and PIDE for option prices. Let the stochas-
tic process X be an admissible time-homogeneous market model with generator

A

V = H 1 ( R

d ) for diffusion market mod-

and let be g be sufficiently smooth and

H m (

d ) , m

( 0 , 1 ) d els, V = R =[ Y 1 / 2 ,...,Y d / 2 ]∈ for general space and time-

H m (x) (

d ) as in Definition 16.2.2 , with m (x)

homogeneous models and V = R =

[ , for time-homogeneous admissible market models con-

sidered here. Then, we obtain formally from semigroup theory and from the rep-
resentation u(t, x) = T t (g) = e − r(T − t)

Y 1 (x 1 )/ 2 ,...,Y d (x d )/ 2 ]

E[ g(X t ) | X 0 = x ] by differentiation in t the

following backward PIDE