of a counting process is a measure of the rate of change of its predictable part.
If a stochastic process
is a counting process, then it is a submartingale, and in particular its Doob-Meyer decomposition is where
is a predictable increasing process.
is called the cumulative intensity of
λ
by Given probability space
and a counting process
which is adapted to the filtration
{ λ ( t ) , t ≥ 0 }
defined by the following limit: The right-continuity property of counting processes allows us to take this limit from the right.
In statistical learning, the variation between
λ
λ ^
can be bounded with the use of oracle inequalities.
If a counting process
copies are observed on that interval,
, then the least squares functional for the intensity is which involves an Ito integral.
If the assumption is made that
is piecewise constant on
, i.e. it depends on a vector of constants
so that they are orthonormal under the standard
norm, then by choosing appropriate data-driven weights
which depend on a parameter
and introducing the weighted norm the estimator for
With these preliminaries, an oracle inequality bounding the
, with probability greater than or equal to