Expectation Discounted Dividend Function For A Class Of Sparse Models

In the dual risk model, gain is the main way for a company that specializes in inventions and discoveries. Therefore, the dividend-payment strategy of dual risk model has become one of the current topics.We always assume that expense is a linear function in the dual risk model.And the expense arriving number process is independent with the gain arriving number process. Actually,it is not enough to fully describe the situation. So the more dependent model is studied.We introduce barrier strategy and threshold strategy to the model on the basis of the dual risk model and generalize to more general case which expense number process is no longer a constant but a random variable.We assume the expense arriving number pro-cess N(t) is a Poisson process with parameterλ> OMeantime,the gain number arriving process Np(t) is a thinning process of the expense arriving number process.We obtain the integral differential equation of the expected discounted dividend function under differ-ent dividend strategy, and we also gives the integral differential equation of the expected discounted penalty function under barriers dividend strategy.Finnally,we obtain solution to the integral differential equation when expense and profit sizes are exponentially dis-tributed.This paper is divided into four chapter:In chapter one,we introduce the current research situation of dual risk model,random premium and dividend problem.We also introduce the dependent risk modelIn chapter two, firstly, we derive an integral equation for the expected discounted dividend payments with the barrier dividend into the model.when expense and profit sizes are exponentially distributed, V(u; b)=∑4i=1CieζiIn chapter three, at first, we derive an integral equation for the expected discounted dividend until ruin under the threshold dividend strategy.0≤u<b V(u; b) satisfied u≥b, V(u; b) satisfiedIn chapter four, we derive an integral equation for the expected penalty dividend under the barrier dividend strategyσ=0,when expense and profit sizes are exponentially distributed,the solution of the time of ruin ψb(u)...