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 $\{N(t),t\geq 0\}$ 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

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

Definition

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

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

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

Estimation

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

If a counting process $N(t)$ is restricted to $t\in [0,1]$ 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

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

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

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

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

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

the estimator for $\beta$ can be given:

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

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

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

with probability greater than or equal to $1-12.85e^{-x}$.