| With the development of modern market economy, market participants have astrong increasing demand for risk control subjects, especially in the insurance industry.Risk control issues will determine the long-term safety of an insurance company’sability to run down. Risk theory becomes more and more popular as a topic ofdiscussion and research，because of its foundation status.For an insurance company, the operations of their company can be analyzed fromthe company’s assets surplus situation, so the accuracy of the surplus assets model,and whether or not the operation is convenient，have an important relationship with acompany the long-term survival and development.As the core of risk theory, ruin theory is becoming the center of attention inthe field of both mathematical statistics and financial mathematics，and is impr-oving better and better.From the past until now，Surplus model of various kinds are put forward. Wewant to obtain the more perfect theory and practical. With the replacement ofknowledge and the demand of realistic. During the process, there are fourclassic and representative model theories have been put forward. And some researchresults have been found out in this field.Cramer-Lundberg’s classical risk model is the most basic in this respect.Defines the basic assets surplus model:U(t)=A t L t,t0.where A t stands for the assets of the momentt. A t stands for the liabilities ofthe momentt. U t stands for the surplus.The liabilities just consider of claims, and the assets just consider the initialsurplus u and premium income, ignoring the influence of the interest rates, theinflation, the dividend, the operation fee etc. So the assets surplus model now can be written: Define Ψ(u,t) is the ruin probability from0to t. Thus, the ruin probability satisfies the defective renewal equation: Then with the research by Hans U. Gerber, Elias S.W. Shiu and Gordon E. Willmot etc. The system formed which use Gerber-Shiu as a tool to solve the ruin theory, considered as a whole of the surplus before ruin, the deficit at ruin, the time of ruin, and ruin probability: And it satisfies the equation:When the influence of interest rates are taken into account, it comes to two parts risk model issue.(insurance risk and financial risk) Follows: Taken in transition probabilities: Satisfies the equation under the model:Then combined with the jump process of claims, we can attribute the diversification of the surplus to three sources. So it’s a particular model, the jump diffusion model, under which we consider the absolute ruin probability. And it fits: Separate study and comparative observation to above models, we will find that there will be some limits of the issues, because of there narrow conditions. While our new model can substantially improve the situation. Furthermore, conclusions can be proven explicitly with the knowledge of financial stochastic analysis.Our new model is given: By the transformation, further rewritten as: where andIn the article, we focus on the situation of σ2=0ruin probability satisfies the equation. Now the model can be simplified as:Get the following conclusions:Theoreml the above formula has the solution:Theorem2let θ be constant, λ1,…,λM be positive constants, define measure transformation as follow: Then transform the model to risk-neutral measure, turns: where is a Brownian motion, is a martingale.Theorem3With the above model under risk-neutral measure, transform probab-ility satisfies the differential equation:Theorem4The crossing probability satisfies the differential equation:Afterwards, we give a discussion to degenerate case about ruin probability. Consistent with the previous model proves our model’s rationality and practicality, conversely. Simultaneously, this model is also raised to broaden the direction of research and development in this field, which is very valuable models and methods. |
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