Erlang (n) Surplus Process With Debit Interest And Dividend Payments

This thesis, mainly to introduce absolute ruin and dividend strategies intothe Erlang(n) surplus process. The entire thesis are divided into 6 chapters.In Chapter 1, we introduce three interested topics, the Erlang(n) surplusprocess, absolute ruin and dividend strategies, in risk theory literature and re-view some corresponding recent research developments in its literature. Then,we set a general framework for the entire paper.In Chapter 2, an expected discounted penalty function at the absolute ruintime is defined for the Erlang(n) surplus process with debit interest. First, wederive a system of integro-differential equations satisfied by the function andobtain a defective renewal equation which links the integro-differential equa-tions. Then, when the initial surplus goes to infinity, the asymptotic behaviorof the absolute ruin probability both for light-tailed and heavy-tailed claims isderived. Finally, closed-form expression for the function for n = 2 and expo-nentially distributed claim amounts is obtained.Chapter 3 and Chapter 4 are dedicated to the study the Erlang(n) surplusprocess with constant barrier strategy and threshold dividend strategy underabsolute ruin, respectively. Systems of integro-differential equations are de-rived for the expected discounted penalty function, the moment generatingfunction and the m-th moment of the discounted dividend payments prior toabsolute ruin. We explicitly solve the three functions for n = 2 and exponen-tially distributed claim amounts, for each model.In Chapter 5, for the classical compound Poisson surplus process, i.e. n =1 in Erlang(n) surplus process, we found that the optimal strategy must be athreshold dividend strategy and showed how the optimal level of the barrierbe determined.In Chapter 6, we introduce the Erlang(n) surplus process with linear div-idend strategy under absolute ruin and its main problems. But we left those problems for it is difficult to solve variable coefficient partial differential equa-tions.