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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.

91B30 Risk theory, insurance (MSC2010)
Full Text: DOI
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