The Ruin Probability In The Three Risk Models

In the category of the insurance mathematics, which is also called actuarial mathematics, ruin theory is the core content of the risk theory. As the quantity index to evaluate the repayment ability of the insurance company, ruin probability and it's generalization—Gerber-Shiu function stand the important place in ruin theory. In this dissertation, the correlated aggregate claims model, the risk model with a constant dividend barrier and generalized double compound Poisson process are discussed and problems are solved as follows:1. Do a further investigation into the problem of ruin probability φ(u) incorrelated aggregate claims model U(t) = , It generalizes one of theclaims from a compound Poisson process to a generalized Poisson process. Explicit expressions are derived for the ultimate survival probabilities under the assumed model when the claim sizes are exponentially distributed, and a tail equivalencerelationship of φ(u) is proved under the assumption that the claim size isheavy-tailed.2. The Gerber-Shiu discounted penalty function is considered for the risk modelwith a constant dividend barrier U(t) = u + ct-S(t), S(t) = . The intergrodifferential equation is discussed and solved when claim occurrence relates to Erlang(2) process. The mathematically tractable formulas are discussed for the Gerber-Shiu function in the situation where the time until the first claim is specially distributed for a class of delayed renewal risk processes with barrier.3. Generalize the generalized double Poisson process to the new model where thetwo claim processes are both the generalized Poisson process R(t)= u + S(t), S(t) = . Then the Lundberg inequality and the commonformula of the ruin probabilities are gotten in terms of some techniques formmartingale theory. Finally the explicit formula of the ruin probabilities is gotten when the two claim distributions are both exponentially distributed.