Page 102 - Special Topic Session (STS) - Volume 3

STS517 Jan Rosinski´
has the same distribution as Y. Y will be called a canonical spectral
̃
representation of Y.
Notice that this is a non-linear stochastic integral but is well defined due to
condition (L1) of Definition 1 (see Rosinski, J. (2018) for details.).
Finally, we should mention transfer of regularity for Lévy measures. In
short, this property says that path regularities of infinitely divisible processes
are inherited by supports of their path Lévy measures. A precise statement
follows (cf. Rosinski, J. (2018)).
t t∈T be an infinitely divisible
Theorem 4 (Transfer of regularity). Let Y = (Y )
process with a σ-finite Lévy measure ν. Assume that paths of Y lie in a set U
that is a standard Borel space for the σ-algebra = ∩ U and also that U
T
is an algebraic subgroup of ℝ under addition. Then ν is concentrated on U in
T
the sense that ν ∗ (ℝ \U) = 0. Therefore, both ℒ(X) and its Lévy measure ν
T
are carried by U.
This implies, for instance, that the Lévy measure of an infinitely divisible
process on = [0,1] having twice continuously differentiable sample paths
must be concentrated on [0,1].
2

3. Result
Now we are ready state the results.
Theorem 5. Let X = (X ) be a Poissonian infinitely divisible process with a
t t∈T
σ-finite Lévy measure ν and given by its canonical spectral representation

t t∈T be an
where N is a Poisson random measure with intensity ν. Let Z = (Z )
dℒ(Z)
arbitrary process independent of N such that ℒ(Z) ≪ v. Put q ≔ .
dv
Then for any measurable functional F ∶ ℝ ↦ ℝ
T

where

t t∈T be a Poissonian infinitely divisible process with a
Theorem 6. Let X = (X )
σ-finite Lévy measure ν and given by its canonical spectral representation

(j)

where N is a Poisson random measure with intensity ν. Let Z (j) = (Z ) t∈T be
t
independent processes that are also independent of N such that ℒ(Z ) ≪ v .
(j)
Then for any measurable functional F ∶ ℝ ↦ ℝ
T

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