| Definition 2.1.1 |
| Written by theoretic |
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A stochastic process N = (N(t))t>o is said to be a Poisson process if the following conditions hold: (1) The process starts at zero: N(0) = 0 a.s. (2) The process has independent increments: for any ti, i = 0,...,n, and n > 1 such that 0 = to < t\ < < tn, the increments N(ti-i,ti], i = 1,. .., n, are mutually independent. (3) There exists a non-decreasing right-continuous function /л : [0, oo) —> [0, oo) with /x(0) = 0 such that the increments N(s,t] for 0 < s < t < oo have a Poisson distribution Pois(/i(s, £]). We call /л the mean value function of N. (4) With probability 1, the sample paths (N(t,u>))t>o of the process N are right-continuous for t > 0 and have limits from the left for t > 0. We say that N has cadlag (continue a droite, limites a gauche) sample paths. We continue with some comments on this definition and some immediate consequences. We know that a Poisson random variable M has the rare property that A = EM = var(M), i.e., it is determined only by its mean value (= variance) if the distribution is specified as Poisson. The definition of the Poisson process essentially says that, in order to determine the distribution of the Poisson process N, it suffices to know its mean value function. The mean value function /л can be considered as an inner clock or operational time of the counting process N. Depending on the magnitude of n(s,t] in the interval (s,i], s < t, it determines how large the random increment N(s,t] is. Since N(0) = 0 a.s. and /x(0) = 0, N(t) = N(t) — N(0) = N(0,t] ~ Pois(/i(0,£]) = Pois(/x(t)). We know that the distribution of a stochastic process (in the sense of Kolmogorov’s consistency theorem1) is determined by its finite-dimensional distributions. The finite-dimensional distributions of a Poisson process have a rather simple structure: for 0 = to < ti < < tn < oo, (N(ti), N(o), • • •, N(tn)) = n ^ N(t1),N(t1) + N(t1,t2]N(t1)+N(t1,t2]+N(t2,t3]---,Y,N(ti-1^]- i=i where any of the random variables on the right-hand side is Poisson distributed. The independent increment property makes it easy to work with the finite-dimensional distributions of N: for any integers ki > 0, i = 1,..., n,P(N(ti) = k\,N(o) = k\ + k2 ,...,N(tn) =&! + ••• + kn) = P(N(ti) = k\,N(ti,t2] = k2 , ,N(tn-i,tn] = kn) 7! 7*** 7! «i «2! kn -u(tn) (M(^l))fcl (/"(^b^])^ (M(^n-l^n])fcrl k\! k^! kn! The cadlag property is nothing but a standardization property and of purely mathematical interest which, among other things, ensures the measur-ability property of the stochastic process N in certain function spaces.2 As a matter of fact, it is possible to show that one can define a process N on [0, 00) satisfying properties (1)-(3) of the Poisson process and having sample paths which are left-continuous and have limits from the right.3 Later, in Section 2.1.4, we will give a constructive definition of the Poisson process. That version will automatically be cadlag. |