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 $\{N(t),t\ge 0\}$ is a counting process, then it is a submartingale, and in particular its Doob-Meyer decomposition is

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

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

$\mathrm{\Lambda }(t)=\underset{0}{\overset{t}{\int }}\lambda (s)ds$.

Definition

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

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

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, $N_1,N_2,\dots ,N_n$, then the least squares functional for the intensity is

$R_n(\lambda )=\underset{0}{\overset{1}{\int }}\lambda (t{)}^{2}dt-\frac{2}{n}\sum _{i=1}^n\underset{0}{\overset{1}{\int }}\lambda (t)d{N}_i(t)$

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 $\beta =({\beta }_1,{\beta }_2,\dots ,{\beta }_m)\in {\mathbb{ℝ}}_{+}^m$ 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

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

the estimator for $\beta$ can be given:

$\hat{\beta }=\arg \underset{\beta \in {\mathbb{ℝ}}_{+}^m}{\min }\{R_n({\lambda }_{\beta })+\Vert \beta {\Vert }_{\hat{w}}\}$.

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

$\Vert \hat{\lambda }-\lambda {\Vert }^2\le \underset{\beta \in {\mathbb{ℝ}}_{+}^m}{\inf }\{\Vert {\lambda }_{\beta }-\lambda {\Vert }^2+2\Vert \beta {\Vert }_{\hat{w}}\}$

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