Interval estimates of the classical ruin probability for some classes of distributions of claims. (English. Ukrainian original) Zbl 0987.91041

Kartashov, M. V.

Theory Probab. Math. Stat. 63, 87-97 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 80-89 (2000).

Let us consider the risk process \(X_{t}=ct-\sum_{s=1}^{N(t)}Z_{s}\), where \(\{Z_{s}\), \(s\geq 1\}\) is a sequence of independent identically distributed non-negative random variables with the distribution function \(F(t)=P\{Z_1<t\}\) and the mean \(\mu=EZ_1\); \(N(t)\) is a Poisson process with the intensity \(\lambda\). The ruin probability is defined as \(\psi(t)=P\{u+X_{t}<0\) for some \(t>0\}\), where \(u\geq 0\) is the initial capital of an insurance company. For an arbitrary distribution density \(f\) on \(R_{+}\) let us define the generating function of moments \(\widehat f(u)=\int_{0}^{\infty}\exp(ut)f(t) dt\). We say that a density \(g\) belongs to the class \(M(\gamma)\), if \(\gamma=\inf_{t>0}(g(t)/\int_0^{\infty}g(s) ds)>0\). One of the presented results is the following.
Let the density \(g=(1-F)/\mu\in M(\gamma)\) for some \(\gamma>0\), and let \(\theta\widehat g(\gamma)>1\) (in particular \(\widehat g(\gamma)=\infty\)). Then the equation \(\theta\widehat g(R)=1\) has a unique positive root \(R<\gamma\) and for all \(t\geq 0\) the following interval estimate is true: \[ B_{L}\exp(-Rt)\leq\psi(t)\leq B_{U}\exp(-Rt), \] where \(B_{L}=(\theta-\beta)/(1-\beta)=1-R/\gamma>0\), \[ B_{U}=\psi_{R}(\infty)\theta/(\theta-\beta(1-\psi_{R}(\infty)))<1,\quad \beta=1-(1-\theta)\gamma/R\in (0,\theta), \] \(\psi_{R}(\infty)=\lim_{t\to\infty}\psi(t)\exp(Rt)=(1-\theta)/\theta R{\widehat g}'(R)\).

Reviewer: A.D.Borisenko (Kyïv)

MSC:

91B30

Risk theory, insurance (MSC2010)

Keywords:

interval estimates; classical ruin probability; insurance claims distributions