The intensity 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 martingale and is a predictable increasing process. is called the cumulative intensity of and it is related to by
Given probability space and a counting process which is adapted to the filtration , the intensity of is the process 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 and its estimator can be bounded with the use of oracle inequalities.
If a counting process is restricted to and i.i.d. 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 and can be written
where the have a factor of 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 can be given:
Then, the estimator is just . With these preliminaries, an oracle inequality bounding the norm is as follows: for appropriate choice of ,
with probability greater than or equal to .