Intensity of counting processes explained

The intensity

of a counting process is a measure of the rate of change of its predictable part. If a stochastic process

\{N(t),t\ge0\}

is a counting process, then it is a submartingale, and in particular its Doob-Meyer decomposition is

N(t)=M(t)+Λ(t)

where

M(t)

is a martingale and

Λ(t)

is a predictable increasing process.

Λ(t)

is called the cumulative intensity of

N(t)

and it is related to

Λ(t)=

	t
\int
	0

λ(s)ds

Definition

Given probability space

(\Omega,l{F},P)

and a counting process

\{N(t),t\ge0\}

which is adapted to the filtration

\{l{F}_t,t\ge0\}

, the intensity of

is the process

\{λ(t),t\ge0\}

defined by the following limit:

λ(t)=\lim_h\downarrow

	1
	h

E[N(t+h)-N(t)|l{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

and its estimator

\hat{λ}

can be bounded with the use of oracle inequalities.

If a counting process

N(t)

is restricted to

t\in[0,1]

and

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(λ)=

	1
\int
	0

λ(t)^2dt-

	2
	n

	n
\sum
	i=1

	1
\int
	0

λ(t)dN_i(t)

which involves an Ito integral. If the assumption is made that

λ(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

	m
\R
	+

and can be written

λ_\beta=

	m
\sum
	j=1

\beta_jλ_j,m, λ_j,m=\sqrt{m}

(

j-1

	j
	m

]

where the

λ_j,m

have a factor of

\sqrt{m}

so that they are orthonormal under the standard

L²

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_^m\hat_j|\beta_j - \beta_| ,

the estimator for

\beta

can be given:

\hat{\beta}=

\argmin

\beta\in

	m
\R
	+

\left\{R_n(λ_\beta)+\|\beta\|_\hat{w

}\right\} .

Then, the estimator

\hat{λ}

is just

λ_\hat{\beta

}. With these preliminaries, an oracle inequality bounding the

L²

norm

\|\hat{λ}-λ\|

is as follows: for appropriate choice of

\hat{w}_j(x)

\|\hat{λ}-λ\|²\le

inf

\beta

\in

	m
\R
	+

\left\{\|λ_\beta-λ\|²+2\|\beta\|_\hat{w

} \right\}

with probability greater than or equal to

1-12.85e^-x

.^[2]

Notes and References

Aalen, O. (1978). Nonparametric inference for a family of counting processes. The Annals of Statistics, 6(4):701-726.
http://mzalaya.free.fr/AGG.pdf Alaya, E., S. Gaiffas, and A. Guilloux (2014) Learning the intensity of time events with change-points