| The Poisson Process |
| Written by theoretic |
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In this section we consider the most common claim number process: the Poisson process. It has very desirable theoretical properties. For example, one can derive its finite-dimensional distributions explicitly. The Poisson process has a long tradition in applied probability and stochastic process theory. In his 1903 thesis, Filip Lundberg already exploited it as a model for the claim number process N. Later on in the 1930s, Harald Cramer, the famous Swedish statistician and probabilist, extensively developed collective risk theory by using the total claim amount process S with arrivals Ti which are generated by a Poisson process. For historical reasons, but also since it has very attractive mathematical properties, the Poisson process plays a central role in insurance mathematics. Below we will give a definition of the Poisson process, and for this purpose we now introduce some notation. For any real-valued function / on [0, oo) we write f(s, t] = f(t) — f(s), 0 < s < t < oo . Recall that an integer-valued random variable M is said to have a Poisson distribution with parameter A > 0 (M ~ Pois(A)) if it has distribution _A Afc P(M = k) = e ! , к = 0,1,... . k We say that the random variable M = 0 a.s. has a Pois(0) distribution. Now we are ready to define the Poisson process. |