Estimate for probability of ruin of insurance company for some insurance model (Q2739837)

A risk process \(R(t)\) is considered at which the flows of the claims and premiums are described by a generalized Poisson process. The distributions of one claim and one premium are supposed to satisfy the Cramér condition. An upper bound for the ruin probability NEWLINE\[NEWLINE \psi(u)=\lim_{t\to\infty}\Pr\{R(t)+u<0\} NEWLINE\]NEWLINE is derived. E.g. in an example presented at the paper, NEWLINE\[NEWLINE \psi(u)\leq\exp\left\{ -{1\over d}\ln\ln u^{1\over 2b} \left( u+{c\lambda_1-a\lambda_2\over \ln{1\over\nu}} \right) + u^{1/2} \left( {16\over d}\ln^2\ln u^{1\over 2b}+{\lambda_1+\lambda_2\over 2\ln{1\over\nu}} \right) \right\}, NEWLINE\]NEWLINE where \(c\) and \(a\) are expectations of claims and premiums, \(\lambda_1\) and \(\lambda_2\) are intensities of the claims and premiums flows, \(\nu\) is the discount factor, \(b\) and \(d\) are some constants connected with the Cramér condition.