MONOTONICITY AND NON-MONOTONICITY OF DOMAINS OF
STOCHASTIC INTEGRAL OPERATORS
Abstract: A Lévy process on
with distribution
at time
is denoted by
If the improper stochastic integral
of
with respect to
is definable, its distribution is denoted by
The class of all infinitely
divisible distributions
on
such that
is definable is denoted by
The class
its two extensions
and
(compensated and
essential), and its restriction
(absolutely definable) are studied. It is shown that
is monotonic with respect to
which means that
implies
Further,
is monotonic with respect to
but neither
nor
is monotonic with respect to
Furthermore, there exist
,
and
such that
and
An explicit
example for this is related to some properties of a class of martingale Lévy processes.
2000 AMS Mathematics Subject Classification: 60E07, 60G51, 60H05.
Key words and phrases: Improper stochastic integral, infinitely divisible distribution,
Lévy process, martingale Lévy process, monotonic.