Poisson process. Etienne Pardoux. Aix Marseille Université. Etienne Pardoux (AMU) CIMPA, Ziguinchor 1 / 8

Transcription

1 Poisson process Etienne Pardoux Aix Marseille Université Etienne Pardoux (AMU) CIMPA, Ziguinchor 1 / 8

2 The standard Poisson process Let λ > be given. A rate λ Poisson (counting) process is defined as P t = sup{k 1, T k t}, where = T < T 1 < T 2 < < T k < <, the r.v. s {T k T k 1, k 1} being independent and identically distributed, each following the law Exp(λ). We have Proposition For all n 1, < t 1 < t 2 < < t n, the r.v. s P t1, P t2 P t1,..., P tn P tn 1 are independent, and for all 1 k n, P tk P tk 1 Poi[λ(t k t k 1 )]. Etienne Pardoux (AMU) CIMPA, Ziguinchor 2 / 8

3 The standard Poisson process Let λ > be given. A rate λ Poisson (counting) process is defined as P t = sup{k 1, T k t}, where = T < T 1 < T 2 < < T k < <, the r.v. s {T k T k 1, k 1} being independent and identically distributed, each following the law Exp(λ). We have Proposition For all n 1, < t 1 < t 2 < < t n, the r.v. s P t1, P t2 P t1,..., P tn P tn 1 are independent, and for all 1 k n, P tk P tk 1 Poi[λ(t k t k 1 )]. Etienne Pardoux (AMU) CIMPA, Ziguinchor 2 / 8

4 Elementary convergence in law Lemma We have For all n 1, let U n be a B(n, p n ) random variable. If np n λ as n, with λ >, then U n converges in law towards Poi(λ). To do : Exercise Let {P t, t } be a rate λ Poisson process, and {T k, k 1} the random points of this Poisson process, such that for all t >, P t = sup{k 1, T k t}. Let < p < 1. Suppose that each T k is selected with probability p, not selected with probability 1 p, independently of the others. Let P t denote the number of selected points on the interval [, t]. Then {P t, t } is a rate λp Poisson process. Etienne Pardoux (AMU) CIMPA, Ziguinchor 3 / 8

5 Elementary convergence in law Lemma We have For all n 1, let U n be a B(n, p n ) random variable. If np n λ as n, with λ >, then U n converges in law towards Poi(λ). To do : Exercise Let {P t, t } be a rate λ Poisson process, and {T k, k 1} the random points of this Poisson process, such that for all t >, P t = sup{k 1, T k t}. Let < p < 1. Suppose that each T k is selected with probability p, not selected with probability 1 p, independently of the others. Let P t denote the number of selected points on the interval [, t]. Then {P t, t } is a rate λp Poisson process. Etienne Pardoux (AMU) CIMPA, Ziguinchor 3 / 8

6 Rate λ and rate 1 Poisson processes A Poisson process will be called standard if its rate is 1. If P is a standard Poisson process, then {P(λt), t } is a rate λ Poisson process. A rate λ Poisson process (λ > ) is a counting process {Q t, t } such that Q t λt is a martingale. Let {P(t), t } be a standard Poisson process (i.e. with rate 1). Then P(λt) λt is martingale, and it is not hard to show that {P(λt), t } is a rate λ Poisson process. Etienne Pardoux (AMU) CIMPA, Ziguinchor 4 / 8

9 Varying rate Let now {λ(t), t } be a measurable and locally ( integrable ) t R + valued function. Then the process {Q t := P λ(s)ds, t } is called a rate λ(t) Poisson process. Clearly M t = Q t t λ(s)ds is a martingale, i.e. M t is F t = σ{q s, s t} measurable, and for s < t, E[M t F s ] = M s. We now want to consider the case where λ is random. For that purpose, it is convenient to give an alternative definition of the above process Q t. Etienne Pardoux (AMU) CIMPA, Ziguinchor 5 / 8

10 Varying rate Let now {λ(t), t } be a measurable and locally ( integrable ) t R + valued function. Then the process {Q t := P λ(s)ds, t } is called a rate λ(t) Poisson process. Clearly M t = Q t t λ(s)ds is a martingale, i.e. M t is F t = σ{q s, s t} measurable, and for s < t, E[M t F s ] = M s. We now want to consider the case where λ is random. For that purpose, it is convenient to give an alternative definition of the above process Q t. Etienne Pardoux (AMU) CIMPA, Ziguinchor 5 / 8

11 Poisson Random Measures Consider a standard Poisson random measure N on R + 2, which is defined as follows. N is the counting process associated to a random cloud of points in R 2 +. One way to construct that cloud of points is as follows. We can consider R 2 + = i=1 A i, where the A i s are disjoints squares with Lebesgue measure 1. Let K i, i 1 be i.i.d. Poisson r.v. s with mean one. Let {Xj i, j 1, i 1} be independent random points of R2 +, which are such that for any i 1, the Xj i s are uniformly distributed in A i. Then K i N(dx) = δ X i (dx). J i=1 j=1 λ(t) denoting a positive valued measurable function, the above {Q t, t } has the same law as Q t = t λ(s) N(ds, du). Etienne Pardoux (AMU) CIMPA, Ziguinchor 6 / 8

14 Random rate Let now {λ(t), t } be an R + valued stochastic process, which is assumed to be predictable, in the following sense. Let for t F t = σ{n(a), A Borel subset of [, t] R + }, and consider the σ algebra of subset of [, ) Ω generated by the subsets of the form 1 (s,t] 1 F, where s < t and F F s, which is called the predictable σ algebra. We assume moreover that E t λ(s)ds < for all t >. We now define as above Q t = t λ(s) N(ds, du). Etienne Pardoux (AMU) CIMPA, Ziguinchor 7 / 8

17 Lemma We have Q t t λ(s)ds is a martingale. Indication of proof : For any δ >, let Q δ t = t λ(s δ) N(ds, du), where λ(s) = for s <. It is not hard to show that Qt δ t λ(s δ)ds is a martingale which converges in L1 (Ω) to Q t t λ(s)ds. The result follows. If we let σ(t) = inf{r >, r λ(s)ds > t}, we have that P(t) := Q σ(t) is a standard Poisson process, and it is plain that ( t Q t = P ). λ(s)ds Etienne Pardoux (AMU) CIMPA, Ziguinchor 8 / 8