Sur les fermés aléatoires.

*(French)*Zbl 0563.60038
Sémin. de probabilités, XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 397-495 (1985).

[For the entire collection see Zbl 0549.00007.]

This is a detailed investigation of random closed sets \(\omega\) in \(R_+\) of Lebesgue measure zero and of the associated random point processes \(\pi_ g(dt,dx)=\sum_{s\in G}\epsilon_{(s,d_ s- s)}(dt,dx)\) and \(\pi_ d(dt,dx)=\sum_{s\in D}\epsilon_{(s,s-g_ s)}(dt,dx)\), where G(\(\omega)\) (resp. D(\(\omega)\)) are the set of left (resp. right) extremities of the intervals contiguous to \(\omega\), and where \(g=(g_ t;t\geq 0)\) and \(d=(d_ t;t\geq 0))\) are given by \(g_ t(\omega)=\sup \{s;s<t;s\in \omega \}\) and \(d_ t(\omega)=\inf \{s;s>t,s\in \omega \}.\)

After a discussion of a filtration \({\mathcal F}=({\mathcal F}_ t;t\geq 0)\) constructed by means of the \(\sigma\)-field generated by g, the author considers the optional disintegration formula for \(\pi_ g\), i.e. a representation involving local time and Lévy’s kernel associated to the random closed set. He proves that a regenerative set is characterized by the fact that local time is essentially the only additive functional it carries.

In the central part of this article the author investigates an interesting submartingale \(\bar Y=(\bar Y_ t;t\geq 0)\) that admits local time as increasing process in the Doob-Meyer decomposition. A comparison with Tanaka’s formula shows that \(\bar Y\) is the optional projection of \(| B|\) on \({\mathcal F}\), B being Brownian motion. Moreover \(M=(sign B)\bar Y\) is shown to be a martingale with respect to its natural filtration \({\mathcal F}'\). In fact, M is the optional projection of B on \({\mathcal F}'.\)

The last three sections contain a study of some semimartingales associated to Lévy’s kernel (which lead to approximation formulas for local time) and of \(\pi_ d\), especially of its dual predictable projection.

This is a detailed investigation of random closed sets \(\omega\) in \(R_+\) of Lebesgue measure zero and of the associated random point processes \(\pi_ g(dt,dx)=\sum_{s\in G}\epsilon_{(s,d_ s- s)}(dt,dx)\) and \(\pi_ d(dt,dx)=\sum_{s\in D}\epsilon_{(s,s-g_ s)}(dt,dx)\), where G(\(\omega)\) (resp. D(\(\omega)\)) are the set of left (resp. right) extremities of the intervals contiguous to \(\omega\), and where \(g=(g_ t;t\geq 0)\) and \(d=(d_ t;t\geq 0))\) are given by \(g_ t(\omega)=\sup \{s;s<t;s\in \omega \}\) and \(d_ t(\omega)=\inf \{s;s>t,s\in \omega \}.\)

After a discussion of a filtration \({\mathcal F}=({\mathcal F}_ t;t\geq 0)\) constructed by means of the \(\sigma\)-field generated by g, the author considers the optional disintegration formula for \(\pi_ g\), i.e. a representation involving local time and Lévy’s kernel associated to the random closed set. He proves that a regenerative set is characterized by the fact that local time is essentially the only additive functional it carries.

In the central part of this article the author investigates an interesting submartingale \(\bar Y=(\bar Y_ t;t\geq 0)\) that admits local time as increasing process in the Doob-Meyer decomposition. A comparison with Tanaka’s formula shows that \(\bar Y\) is the optional projection of \(| B|\) on \({\mathcal F}\), B being Brownian motion. Moreover \(M=(sign B)\bar Y\) is shown to be a martingale with respect to its natural filtration \({\mathcal F}'\). In fact, M is the optional projection of B on \({\mathcal F}'.\)

The last three sections contain a study of some semimartingales associated to Lévy’s kernel (which lead to approximation formulas for local time) and of \(\pi_ d\), especially of its dual predictable projection.

Reviewer: M.Dozzi

##### MSC:

60G07 | General theory of stochastic processes |

60G44 | Martingales with continuous parameter |

60J55 | Local time and additive functionals |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

60G57 | Random measures |