Ruin probability of an insurance company with hyperexponential distribution of insurance premiums and insurance payments for different insurance models

In this paper the calculated formulas which allow to calculate the ruin probability of an insurance company for some insurance models under assumption that insurance premiums and insurance payments have the hyperexponential distribution are obtained. n For Cramer-Lundberg model if the insurance payments distribution is У (S) = I A_ka_keS, rate cash flows is C and the insur- k=1 ance payments flow intensity is X , then the probability of ruin is defined as -Jj P(S) = I Pj j j=1 where у · are the simple positive roots of equation f (z) = C - XI= 0, k=1 _kz n 1 1 and the coefficients P are the solution of equations I -P. =-. j=1 k У j k For Cramer-Lundberg model with stochastic premiums if the insurance premiums and insurance payments distributions are m n Ф(S) = X A_kа_ke, T(S) = X B_kP_ke , respectively, the insurance premiums and insurance payments flows intensities are к=\ к=\ X and Ц , then the probability of ruin is determined by the same relation, but Y j are the simple positive roots of equation f (z) = x£ Bh. __х-ц = о, к=\ _к + к =\ P_k n \ \ and the coefficients P_j are the solution of equations X-Pj = -. j=\ Рк J Рк For Cramer-Lundberg model with MMP payments flow with two states X\ and X₂ and the final probabilities of states P₂ /(P\ + P₂), P\ /(P\ + P₂) the probabilities of ruin 2n PJ (S) = X P^, J = i,2 , к=\ where Yj are the simple positive roots of equation n A а n A а f (z) = (X\ +P\ + Cz -X\X ^-)(Х₂ +P2 + Cz -X2 X P\P 2 = 0, j=\ J J=\ J and the coefficients Pj are the solution of equations 2n а 1 2n а n A __Y n A X-- Рк\ = X-- Yк(C-X\Р = 0, J = i,n; P_k2 = (i + £-(C-X\Х-'-ЪРк к=\ J Yк \ к=\ J Yк ·=\ ,- Yк \ /=\ ,- Yк

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