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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.
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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.
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The Homogeneous Poisson Process, the Intensity Function, the Cramer-Lundberg Model |
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Written by theoretic
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The most popular Poisson process corresponds to the case of a linear mean value function /л:
/x(t) = At, t > 0 ,
for some A > 0. A process with such a mean value function is said to be homogeneous, inhomogeneous otherwise. The quantity A is the intensity or rate of the homogeneous Poisson process. If A = 1, N is called standard homogeneous Poisson process.
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A homogeneous Poisson process with intensity |
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Written by theoretic
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A homogeneous Poisson process with intensity A has
(1) cadlag sample paths,
(2) starts at zero,
(3) has independent and stationary increments,
(4) N(t) is Pois(At) distributed for every t > 0.
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Written by theoretic
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Finance and insurance have been merging for many years. Among other things, insurance companies invest in financial derivatives (options, futures, etc.) which are commonly modeled by functions of Brownian motion such as solutions to stochastic differential equations. If one wants to take into account jump characteristics of real-life financial/insurance phenomena, the Poisson process, or one of its many modifications, in combination with Brownian motion, offers the opportunity to model financial/insurance data more realistically. In this course, we follow the classical tradition of non-life insurance, where Brownian motion plays a less prominent role.
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The Cramer-Lundberg model |
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Written by theoretic
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The homogeneous Poisson process plays a major role in insurance mathematics. If we specify the claim number process as a homogeneous Poisson process, the resulting model which combines claim sizes and claim arrivals is called Cramer-Lundberg model :
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Written by theoretic
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Poisson processes constitute one particular class of Markov processes on [0, 00) with state space No = {0,1,...}. This is a simple consequence of the independent increment property. It is left as an exercise to verify the Markov property, i.e., for any 0 = to < t\ < • • • < tn and non-decreasing natural numbers ki > 0, i = 1,...,n, n > 2,
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Written by theoretic
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(Relation of the intensity function of the Poisson process and its Markov intensities)
Consider a Poisson process N = (N(t))t>o which has a continuous intensity function A on [0, oo). Then, for к > 0,
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Relations Between the Homogeneous and the Inhomogeneous Poisson Process |
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Written by theoretic
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The homogeneous and the inhomogeneous Poisson processes are very closely related: we will show in this section that a deterministic time change transforms a homogeneous Poisson process into an inhomogeneous Poisson process, and vice versa.
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