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Recursive moments of compound renewal sums with discounted claims. (English) Zbl 0979.91048
The authors consider the following risk model. (i) The claim counting process \(N(t)\) forms an ordinary renewal process: the positive claim occurrence times are given by \(\{T_{k}\}_{k\geq 1}\); the i.i.d. positive inter-occurrence times \(\tau_{k}=T_{k}-T_{k-1}\), \(k\geq 2\) and \(\tau_1=T_1\) have a common continuous distribution function \(F\); the Laplace transform \(L_t\) of \(T_1\) exists over a subset of \(R\). (ii) The corresponding deflated claim severities \(\{X_{k}\}_{k\geq 1}\) are such that \(\{X_{k}\}_{k\geq 1}\) are i.i.d.; \(\{X_{k},\tau_{k}\}_{k\geq 1}\) are mutually independent; the moment generating function \(M_X\) of \(X_1\) exists over a subset of \(R\), \(\mu_{k}=E(X_1^{k})>0\). (iii) The aggregate discounted value at time \(0\) of the inflated claims recorded over the period \([0,t]\) yields \(Z(t)=\sum_{k=1}^{N(t)}e^{-\delta T_{k}}X_{k}\), with \(Z(t)=0\) if \(N(t)=0\). Here \(\delta\) is the net interest rate, acting on claim severities at time \(t\). The authors obtain such recursive formulas \[ \begin{aligned} M_{Z(t)}^{(n)}(0) &= \sum_{k=0}^{n-1}{n \choose k}\mu_{n-k}\int_{0}^{t}e^{-n\delta v}M_{Z(t-v)}^{(k)}(0) dm(v),\\ M_{Z(\infty)}^{(n)}(0) &= {L_{T}(n\delta)\over 1-L_{T}(n\delta)}\sum_{k=0}^{n-1}{n \choose k}\mu_{n-k}M_{Z(\infty)}^{(k)}(0) \end{aligned} \] where \(m(t)=E[N(t)]\), \(M_{Z(t)}^{(n)}(s)\) is the \(n\)-th order derivative at \(s\) of \(M_{Z(t)}(s)=E[e^{sZ(t)}]\). Two examples – mixed Poisson process and Erlang process as well as some numerical illustrations are presented. The generalizations to delayed and stationary renewal processes are discussed.

MSC:
91B30 Risk theory, insurance (MSC2010)
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