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

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 $$B_{L}=(\theta-\beta)/(1-\beta)=1-R/\gamma>0$$, $B_{U}=\psi_{R}(\infty)\theta/(\theta-\beta(1-\psi_{R}(\infty)))<1,\quad \beta=1-(1-\theta)\gamma/R\in (0,\theta),$ $$\psi_{R}(\infty)=\lim_{t\to\infty}\psi(t)\exp(Rt)=(1-\theta)/\theta R{\widehat g}'(R)$$.

### MSC:

 91B30 Risk theory, insurance (MSC2010)