Intensity of counting processes

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

.

Definition

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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.[1]


Estimation

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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  .[2]

References

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  1. ^ Aalen, O. (1978). Nonparametric inference for a family of counting processes. The Annals of Statistics, 6(4):701-726.
  2. ^ Alaya, E., S. Gaiffas, and A. Guilloux (2014) Learning the intensity of time events with change-points[permanent dead link]