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As an important part of the actuarial risk theory, mainly used in the financial, insurance, securities, investment and risk management in the literature [28] compound Poisson-Geometric process on the basis of the proposed multi-insurance model study, the main addresses the following issues: 1. study a class of multi-insurance model, U (t) uin Kj i1 (t) Yij in Nj i1 (t) XijW (t) = {N i is the company's initial surplus (t ); t ≥ 0} are parameters λi, ρi compound Poisson-Geometric process, premiums came to the number of {K i (t); t ≥ 0} is the number of parameters GG for αi the Poisson process, i = 1, L, n, {W (t), t ≥ 0} is a standard Wiener process, σ is the perturbation strength. adjustment coefficient for this model as well as the bankruptcy probability expressions and degradation for the case of double insurance of Gerber -Shiu discounted penalty function .2. constant interest rate model, the first model for dUδ (t) = Uδ (the t) δdt cdM-(t)? dS (t) {M (t), t ≥ 0) } a the parameters λ1 The Poisson process ΣS (t) = iN = 1 (t) Xi, claims the arrival process {N (t, t ≥ 0)} is the parameter for λ2 the ρ compound Poisson-Geometric process, interest force constant δ and δ gt; 0 has been integral equation satisfied by the probability of survival, the purpose of correction of the literature [39] in the derivation of error; another model for ∫ Uδ (t) = ueδt cst (δ)? 0t eδ (t? x) dS (t), ΣΣ == 11 () (1) = 21 () (2) S (t) iN tXi Nj tXj, {N i (t); t ≥ 0} (i = 1,2) are parameters λi, ρi compound Poisson-Geometric process, given the model's initial assets to meet u survival probability integral equation and initial assets exact solution when the survival probability .3 in the literature [28] model based on premium income of promotion to the Markov environment, research the following model ∫ ΣU (t) = u 0t cI the ds? iN = 1 ( t) Xis wherein, {I t} T ≥ 0 is a finite state stationary ergodic Markov jump process, wherein premium rate is under the Markov jump process environment, i.e. the rates of the time t for c It, When It is in the state i, the rate constants ci, i = 1,2, L, n for a given initial state of the Markov process, determined the conditions satisfied by the ruin probability integral equation and derivation of a the ruin probability recursive Markov processes with a smooth initial distribution inequality .4. study a class of double insurance model, the model follows ΣΣ =? = 11 () (1)? = 21 () (2) U (t) uctiN tXi Nj tXj wherein {N 1 (t); t ≥ 0} parameter is λ, ρ composite Poisson-Geometric process {N 2 (t); T ≥ 0} is an update process their to the time interval of {V i} i ≥ 1, is assumed here that the {V i} i ≥ 1 separate same obedience generalized Erlang (n) distribution parameter is λ1 of λ2, and L, and λn, i.e. Vi can be decomposed into Vi = Vi1 Vi2 L Vin, where V ij obey parameter as λj exponential distribution. Gerber-Shiu discounted penalty function is decomposed into two parts, the Gerber-Shiu discounted penalty function satisfied integral equation, using the martingale method to get the the model the Lundberg equations, and Laplace transform gives the exact solution of the initial capital for the 0:00 Gerber-Shiu discounted penalty letter.
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