Some generalization of the ruin probability problem in the classical risk theory (Q2740075)

The author considers the following generalization of the ruin probability problem in the classical risk theory NEWLINE\[NEWLINE {\gamma_1+\lambda_1 e^{-u}\over \gamma_2+\lambda_2 e^{-u}}\varphi'(u)=\varphi(u)-\int_0^{u}\varphi(u-t) dF,\quad u\geq 0, NEWLINE\]NEWLINE where \(\lambda_1,\lambda_2>0\), \(\gamma_1, \gamma_2>0\), \(\gamma_1\lambda_2-\gamma_2\lambda_1\neq 0\), \(F(t)\) is a distribution function of individual claim amount, \(u\) is the initial capital of the insurance company, \(\varphi(u)\) is the probability of no ruin, \(0\leq\varphi(u)\leq 1\), \(\varphi(\infty)=1\). Under some conditions the existence of a solution of the considered problem is proved.