Intensity of counting processes

The intensity ${\displaystyle \lambda }$ of a counting process is a measure of the rate of change of its predictable part. If a stochastic process ${\displaystyle \{N(t),t\geq 0\}}$ is a counting process, then it is a submartingale, and in particular its Doob-Meyer decomposition is

${\displaystyle N(t)=M(t)+\Lambda (t)}$

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

${\displaystyle \Lambda (t)=\int _{0}^{t}\lambda (s)ds}$.

Definition

Given probability space ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$ and a counting process ${\displaystyle \{N(t),t\geq 0\}}$ which is adapted to the filtration ${\displaystyle \{{\mathcal {F}}_{t},t\geq 0\}}$, the intensity of ${\displaystyle N}$ is the process ${\displaystyle \{\lambda (t),t\geq 0\}}$ defined by the following limit:

${\displaystyle \lambda (t)=\lim _{h\downarrow 0}{\frac {1}{h}}\mathbb {E} [N(t+h)-N(t)|{\mathcal {F}}_{t}]}$.

The right-continuity property of counting processes allows us to take this limit from the right.^[1]

Estimation

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

If a counting process ${\displaystyle N(t)}$ is restricted to ${\displaystyle t\in [0,1]}$ and ${\displaystyle n}$ i.i.d. copies are observed on that interval, ${\displaystyle N_{1},N_{2},\ldots ,N_{n}}$, then the least squares functional for the intensity is

${\displaystyle R_{n}(\lambda )=\int _{0}^{1}\lambda (t)^{2}dt-{\frac {2}{n}}\sum _{i=1}^{n}\int _{0}^{1}\lambda (t)dN_{i}(t)}$

which involves an Ito integral. If the assumption is made that ${\displaystyle \lambda (t)}$ is piecewise constant on ${\displaystyle [0,1]}$, i.e. it depends on a vector of constants ${\displaystyle \beta =(\beta _{1},\beta _{2},\ldots ,\beta _{m})\in \mathbb {R} _{+}^{m}}$ and can be written

${\displaystyle \lambda _{\beta }=\sum _{j=1}^{m}\beta _{j}\lambda _{j,m},\;\;\;\;\;\;\lambda _{j,m}={\sqrt {m}}\mathbf {1} _{({\frac {j-1}{m}},{\frac {j}{m}}]}}$,

where the ${\displaystyle \lambda _{j,m}}$ have a factor of ${\displaystyle {\sqrt {m}}}$ so that they are orthonormal under the standard ${\displaystyle L^{2}}$ norm, then by choosing appropriate data-driven weights ${\displaystyle {\hat {w}}_{j}}$ which depend on a parameter ${\displaystyle x>0}$ and introducing the weighted norm

${\displaystyle \|\beta \|_{\hat {w}}=\sum _{j=2}^{m}{\hat {w}}_{j}|\beta _{j}-\beta _{j-1}|}$,

the estimator for ${\displaystyle \beta }$ can be given:

${\displaystyle {\hat {\beta }}=\arg \min _{\beta \in \mathbb {R} _{+}^{m}}\left\{R_{n}(\lambda _{\beta })+\|\beta \|_{\hat {w}}\right\}}$.

Then, the estimator ${\displaystyle {\hat {\lambda }}}$ is just ${\displaystyle \lambda _{\hat {\beta }}}$. With these preliminaries, an oracle inequality bounding the ${\displaystyle L^{2}}$ norm ${\displaystyle \|{\hat {\lambda }}-\lambda \|}$ is as follows: for appropriate choice of ${\displaystyle {\hat {w}}_{j}(x)}$,

${\displaystyle \|{\hat {\lambda }}-\lambda \|^{2}\leq \inf _{\beta \in \mathbb {R} _{+}^{m}}\left\{\|\lambda _{\beta }-\lambda \|^{2}+2\|\beta \|_{\hat {w}}\right\}}$

with probability greater than or equal to ${\displaystyle 1-12.85e^{-x}}$.^[2]

References

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]}