| The claim number process |
| Written by theoretic |
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The iid property of the claim sizes, Xi, reflects the fact that there is a homogeneous probabilistic structure in the portfolio. The assumption that claim sizes and claim times be independent is very natural from an intuitive point of view. But the independence of claim sizes and claim arrivals also makes the life of the mathematician much easier, i.e., this assumption is made for mathematical convenience and tractability of the model. Now we can define the claim number process N(t) = #{i > 1 : Ti < t} , t > 0, i.e., N = (N(t))t>o is a counting process on [0, 00): N(t) is the number ofthe claims which occurred by time t. The object of main interest from the point of view of an insurance company is the total claim amount process or aggregate claim amount process:1 N(t) oo s(t)= 2_. Xi = 2_. Xi /[0,4] (Ti), t > 0. The process S = (S(t))t≥0 is a random partial sum process which refers to the fact that the deterministic index n of the partial sums Sn = X1 + • • • + Xn is replaced by the random variables N(t): S(t) = X1 +•••+XN(t) =SN(t) , t ≥0. It is also often called a compound (sum) process. We will observe that the total claim amount process S shares various properties with the partial sum process. For example, asymptotic properties such as the central limit theorem and the strong law of large numbers are analogous for the two processes; see Section 3.1.2. |