Renewal Equation And The Ruin Probability With Asymptotic Independent Insurance And Financial Risks

Risk models are divided to continuous and discrete model. Classical risk theorymost studies on ruin probability, and focus on ruin probability. Then, many actuarialexperts induct the deficit at ruin and the surplus immediately before ruin, and get lots ofbeautiful results.Firstly, we study the renewal equation of L-C risk model. The renewal equation isthe core equation of ruin probability and usually induced by mathematical analysis. Weconsider the L-C model and model with constant interest, and give the new method ofinducing renewal equation by the ruin functions. The regularization of surplusimmediately before ruin is ruin probability. And the regularization of the deficit at ruinand the surplus immediately before ruin are independent on the case of an exponentialindividual claim amount distribution.Secondly, the present paper considers a kind of dependent discrete risk model,which is under both insurance and financial risks. Usually, this model is considered withindependent insurance and financial risks because of the difficulty in solving theproduct of two dependent random variables. We introduce asymptotic independence todescribe dependence with is used in limit theory. So, we use a new definition ofasymptotic independence to study conduct of two random variables. Then, theverification of asymptotic independence is studied by copula theory and give threeexamples: AMK， FGM and Frank copulas. When the insurance risk followsextended-regular-varying and regular varying tail, we derive an asymptotic formula ofthe finite and infinite ruin probability.