Intensity of counting processes

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

[math]\displaystyle{ N(t) = M(t) + \Lambda(t) }[/math]

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

[math]\displaystyle{ \Lambda(t) = \int_{0}^{t} \lambda(s)ds }[/math].

Definition

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

[math]\displaystyle{ \lambda(t) = \lim_{h\downarrow 0} \frac{1}{h} \mathbb{E}[N(t+h) - N(t) | \mathcal{F}_t] }[/math].

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

Estimation

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

If a counting process [math]\displaystyle{ N(t) }[/math] is restricted to [math]\displaystyle{ t\in [0,1] }[/math] and [math]\displaystyle{ n }[/math] i.i.d. copies are observed on that interval, [math]\displaystyle{ N_1, N_2, \ldots, N_n }[/math], then the least squares functional for the intensity is

[math]\displaystyle{ R_n(\lambda) = \int_{0}^{1} \lambda(t)^2dt - \frac{2}{n} \sum_{i=1}^n \int_{0}^{1}\lambda(t)dN_i(t) }[/math]

which involves an Ito integral. If the assumption is made that [math]\displaystyle{ \lambda(t) }[/math] is piecewise constant on [math]\displaystyle{ [0,1] }[/math], i.e. it depends on a vector of constants [math]\displaystyle{ \beta = (\beta_1, \beta_2, \ldots, \beta_m) \in \R_+^m }[/math] and can be written

[math]\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}]} }[/math],

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

[math]\displaystyle{ \|\beta\|_{\hat{w}} = \sum_{j=2}^m\hat{w}_j|\beta_j - \beta_{j-1}| }[/math],

the estimator for [math]\displaystyle{ \beta }[/math] can be given:

[math]\displaystyle{ \hat{\beta} = \arg\min_{\beta\in \R_+^m} \left\{R_n(\lambda_\beta) + \|\beta\|_{\hat{w}}\right\} }[/math].

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

[math]\displaystyle{ \|\hat{\lambda} - \lambda\|^2 \le \inf_{\beta \in \R_+^m} \left\{ \|\lambda_\beta - \lambda\|^2 + 2\|\beta\|_{\hat{w}} \right\} }[/math]

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