Multidimensional Feller Processes - Computational Methods for Quantitative Finance - page 247

The tuple (c(x), b(x), Q(x), ν(x, d z)) in ( 16.5 ) is called the characteristics of
the Markov process X . We sometimes denote A by − ψ(x,D) . In the following, we

set c(x) 0 for notational convenience and restrict ourselves to a certain kind of

normal Markov processes, the so-called Feller processes ([3, Theorem 3.1.8] states
( 16.1 ) for a Feller process, see also [141, p. 83]). These can be defined via the
semigroup (T t ) t ≥ 0 generated by the corresponding process X . A semigroup (T t ) t ≥ 0
is called Feller if it satisfies

=

d ) , the continuous functions on d

(i) T t maps C ∞ ( R R vanishing at infinity, into

itself:

d ) → C ∞ ( R d )

T t : C ∞ ( R boundedly .

(ii) T t is strongly continuous, i.e. lim t → 0 + u − T t u L ∞ ( R = 0 for all u ∈ d )

d ) .

C ∞ ( R

Spatially homogeneous Feller processes are Lévy processes (cf., e.g. [18, 143]).
Their characteristics, the Lévy characteristics , do not depend on x , see Chaps. 10
and 14.

Example 16.1.2 A standard Brownian motion has the characteristics ( 0 , 1 , 0 ) .An
R -valued Lévy process has characteristics (b,Q,ν( d z)) , for real numbers b , Q ≥ 0

and a jump measure ν with 0 = z ∈R

min ( 1 ,z 2 )ν( d z) < ∞ .

It is interesting to ask which symbols correspond to PDOs that are generators of
Feller processes. This martingale problem is discussed in the following theorem due
to [144].

d d

Theorem 16.1.3 Let ψ be a negative definite symbol , i . e . a mea-

surable and locally bounded function in both variables (x, ξ ) that admits for each
x

: R × R → C

d

∈ R a Lévy-Khinchine representation ( 16.5 ). If

2 ) for all ξ ∈ R d ,

(a) sup x ∈R | ψ(x,ξ) | ≤ κ( 1 + | ξ |

d

(b) ξ → ψ(x,ξ) is uniformly continuous at ξ = 0,

d ,

→ ∈ R

(c) x ψ(x,ξ) is continuous for all ξ

ψ(x,D),C 0

d )) extends to a Feller generator .

then ( − ( R

Remark 16.1.4 We show well-posedness of the pricing equations in Sect. 16.5 .For
the existence of a process with a given symbol, it is sufficient to require (a) from
Theorem 16.1.3 and ψ(x, 0 )

d ,cf.[85, Theorem 3.15], (b) and (c)

= 0 for all x ∈ R

are required to obtain a Feller process.

In the Lévy case, existence of a Lévy process can be proven for any Lévy sym-
bol. This does not hold for Feller processes. For (financial) applications, it is more
convenient to consider the characteristic triplet instead of the symbol. We therefore
make the following assumption on the characteristic triplet in the remainder.