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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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References:
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