Intensity Of Counting Processes

The intensity $\lambda$ 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

:$N(t) = M(t) + \Lambda(t)$

where $M(t)$ is a martingale and $\Lambda(t)$ is a predictable increasing process. $\Lambda(t)$ is called the cumulative intensity of $N(t)$ and it is related to $\lambda$ by

:$\Lambda(t) = \int_^ \lambda(s)ds$.

Definition

Given probability space $(\Omega, \mathcal, \mathbb)$ and a counting process $\$ which is adapted to the filtration $\$, the intensity of $N$ is the process $\$ defined by the following limit:

: \lambda(t) = \lim_ \frac \mathbb \mathcal_t.

The right-continuity property of counting processes allows us to take this limit from the right.

Estimation

In statistical learning, the variation between $\lambda$ and its estimator $\hat$ can be bounded with the use of oracle inequalities.

If a counting process $N(t)$ is restricted to t\in ,1/math> and $n$ i.i.d. copies are observed on that interval, $N_1, N_2, \ldots, N_n$, then the least squares functional for the intensity is

:$R_n(\lambda) = \int_^ \lambda(t)^2dt - \frac \sum_^n \int_^\lambda(t)dN_i(t)$

which involves an Ito integral. If the assumption is made that $\lambda(t)$ is piecewise constant on ,1/math>, i.e. it depends on a vector of constants $\beta = (\beta_1, \beta_2, \ldots, \beta_m) \in \R_+^m$ and can be written

:$\lambda_\beta = \sum_^m \beta_j \lambda_, \;\;\;\;\;\; \lambda_ = \sqrt \mathbf_$,

where the $\lambda_$ have a factor of $\sqrt$ so that they are orthonormal under the standard $L^2$ norm, then by choosing appropriate data-driven weights $\hat_j$ which depend on a parameter $x>0$ and introducing the weighted norm

:$\, \beta\, _ = \sum_^m\hat_j, \beta_j - \beta_,$,

the estimator for $\beta$ can be given:

:$\hat = \arg\min_ \left\$.

Then, the estimator $\hat$ is just $\lambda_$. With these preliminaries, an oracle inequality bounding the $L^2$ norm $\, \hat - \lambda\,$ is as follows: for appropriate choice of $\hat_j(x)$,

: $\, \hat - \lambda\, ^2 \le \inf_ \left\$

with probability greater than or equal to $1-12.85e^$.Alaya, E., S. Gaiffas, and A. Guilloux (2014) Learning the intensity of time events with change-points
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References

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Stochastic processes