On The Phase-type Dual Model Perturbed By Diffusion

In the risk theory,ruin probability and dividend problem have draw lots of atten-tions,while the dual risk model shows that the surplus process has sustained expense and incidental income,which is justified by real-life scenes such as life insurance,etc.Therefore,scholars used to describe some of the securities portfolio,pension business,and enterprises whose income-generating depending on inventions or found etc by the dual model.In recent years,in the insurance actuarial and risk management theory,the dual risk model has been widely concerned.Yang&Sendova(2014)[25]consider a Sparre-Andersen dual risk model and study the ruin probability and dividend prob-lem,then obtaining explicit expression of ruin probability and the integral-differential equation of expectation discounted dividend function.On the basis of Yang&Sendova(2014),Bergel et al.(2016)[5]extend the income process to the Phase-type distribution,and consider the ruin probability and dividend problem.In this paper,we consider a phase-type dual model perturbed by diffusion,whose inter-claim times are continuous phase-type distributed.We derive the integral differ-ential equation and boundary conditions for the ruin probability,and obtain the explicite expression of the ruin probability when the phase type distribution degenerates into the generalized Erlang(n)distribution.Finally,we consider a threshold dividend payment strategy in the model,and we obtain the integral differential equations for the expected dividend discount function.