Sciences & technologie. A, sciences exactes
Volume 0, Numéro 24, Pages 41-45
2006-12-31
Laws Of Excursions Associated To Additive Functionals
Authors : Boutabia H .
Abstract
Let (Pt) be a right borel semigroup and let (St) the right inverse of a continuous additive functional (Bt). Let ( ) Yt t R∈ be a right stationary process with random birth and death, Markov with semi group (Pt) under the Kuznetsov measure Q associated to an excessive measure. We define, under the assumption that the characteristic measure υB = Q I B dt∫01 { }Yt∈. ( ) of (Bt) is purely excessive for the semigroup (Ps), an additive functional for (Yt ) t R∈ in terms of (Bt) and we study the laws of excursions associated to the regenerative set which consists in times of discontinuity of the right inverse (Ut) of this additive functional. More precisely, if we note by (Φ t ) the process ⎛ ⎞⎜ ⎟⎝ ⎠YUt and by H the σ algebra generated by Ht ( )t R∈ where Ht is the Q-completion of Ht0+ ( ⎛ ⎞⎜ ⎟⎝ ⎠Ht0 is the natural filtration of (Φ t ) ), then if T is a ( Ht ) -stopping time such that U UT− ≠ T and ΦT T− ≠ Φ , the conditional law of the excursion straddling ⎤ ⎡⎥ ⎢⎦ ⎣U UT−, T with respect to H depend only on ΦT and ΦT− . Conditional laws of pairs of excursions are also considered
Keywords
Standard process; Predictable process; Excursion; Additive functional; Conditional law; Exit measure; Kuznetsov process. MSC: 60J25; 60J40; 60J55
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