Related Issues In The Gamma-Omega Model

In the classical risk theory, the central problem had been to calculate the probability of ruin of a insurance company. In the same time, a main goal of a company should be the paying of dividends to its shareholders. So the problem of dividends and ruin have become an issue which is concentrated by an increasing proportion of people in recent risk theory.In the traditional actuarial risk model, if the surplus is negative, the company is ruined and has to go out of business. However in same articles which published in recent years, we distinguish between ruin(negative surplus) and bankruptcy (going out of business), it is assumed that even with a negative surplus, the company can do business as usual until bankruptcy occurs. The idea for this notion of bankruptcy comes from the observation that in some industries, companies can continue doing business even though they are technically ruined.In this paper, we main research the Gamma-Omega model. In this model, we assume that the probability of bankruptcy of the company at a point of time is quantified by a bankruptcy rate function ω(x), where x is the value of the negative surplus at that time. At the same time, the dividends of the company can only be paid at certain random times, and thus constitute a discrete sequence of random variables. We assume that the waiting times between successive dates when dividends can be paid are independent random variables with a common exponential distribution with parameter γ.Firstly, we assume that the surplus process of the company is modeled by a wiener process(Brownian motion). By using the strong Markov property, we derive differential equations for the expected discounted dividends until bankruptcy under a threshold strategy, then utilize a auxiliary function, we calculate its formulas.Subsequently, at the same assumption of the surplus process, we also use the strong Markov property, then derive differential equations for the expected discounted value of a penalty at bankruptcy under a barrier strategy, and calculate its explicitly formulas with a constant bankruptcy rate λ. φ(x;b)=κ∫x-∞w(y)eα(x-y)dy+κ∫x0w(y)eβ(x-y)dyFinally,we assume that the surplus process of the company is modeled by a jump which is added to the Brownian motion.In other word,the surplus process is a compound Poisson process perturbed by diffusion.We use the strong Markov property for the third time,then the integral-differential equation for the bannkruptcy probability under a barrier strategy is derived,and its formlas is callculated when a constant bankruptcy rate.