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Sur la charge associée à une mesure aléatoire réelle stationnaire. (French) Zbl 0551.60051

Sémin. probabilités XVIII, 1982/83, Proc., Lect. Notes Math. 1059, 391-401 (1984).
[For the entire collection see Zbl 0527.00020.]
Let \((\theta_ t)\) be a measure preserving, ergodic flow on the probability space (\(\Omega\),\({\mathcal F},P)\) and let N be a random measure on \({\mathbb{R}}\) that is stationary with respect to \((\theta_ t)\). The author shows that if the intensity i (satisfying \(E[N(A)]=i\) \(\times\) (Lebesgue measure of A)) is less than one, then the random variable \(W=\sup \{N((-t,0])-t:t\geq 0\}\) is finite almost surely. (One can interpret W as the ”charge” at time zero, i.e., service demanded but not yet effected, of a unit rate server facing the demand N.) The process \(W_ t=W(\theta_ t)\) is shown to be the unique solution to the stochastic differential equation \(dW_ t+1(W_ t>0)dt+1(W_ t=0)\sigma (\theta_ t)dt=dN_ t.\) where \(\sigma_ t=\sigma (\theta_ t)\) is the absolutely continuous component of N, for which \(\sigma\leq 1\) on \(\{W=0\}\). Several additional properties are established.
Reviewer: A.F.Karr

MSC:

60G57 Random measures
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K25 Queueing theory (aspects of probability theory)

Keywords:

ergodic flow

Citations:

Zbl 0527.00020
Full Text: Numdam EuDML