Several Ruin Problem In Risk Models With Constant Interest Rate

Risk theory is an important part of financial mathematics,and ruin theory is the core of risk theory.More than 100 years ago,Lundberg established the classical risk model,however the conditions of this model were too ideal.Later,many scholars improved and promoted its conditions and conclusions.On the basis of the classical risk model,we consider the joint distribution of the time of ruin and the number of claims until ruin in the classical risk model with interest,and the expected discount penalty function in Markov-modulated risk model with diffusion disturbance and constant interest rate.In the first chapter,the background and development of ruin theory are summarized.In the second chapter,we establish the risk model with constant interest rate and when the initial surplus vu=0,the expression of Gerber-Shiu type function and the probability of ruin at the nth claims through the probabilistic method,and then obtained the probability of ruin when the initial surplus u>0.In the third chapter,we consider the joint density of the time of ruin and the number of claims before and probability of ruin at the nth claim.We obtain the joint distribution of time of ruin and the numbers of total claims before ruin ?n(0,t)when the initial surplus u=0by the conclusion of Dickson,density function of ruin time and expression of Gerber-Shiu function ?r,?(u)by summing?n(0;t)over m,then to ?n(0,t)integral about t to get the ruin probability when initial surplus u =O.Secondly,by classifying and discussing the cases that may lead to ruin,we obtain the integral and differential equation of the joint density ?n(u,t)of the time of ruin Tu and numbers of claim before ruin at the initial surplus u>0 by using the method of probability,and calculate the probability of ruin pn(u)at the nth claim through ?n(u,t).In the four chapter,we consider the moment of the total number of claims until ruin NTu·we obtain the integro-differential equations of Gerber-Shiu function by the method of probability,and when the initial surplus u=0 the first moments of the total number of claims until ruin were obtained by differentiating the Gerber-Shiu type function,and then the first and second moments of the total number of claims until ruin are obtained by solving the inverse Laplace transform of gerber-shiu type function ?r(u).In the fifth chapter,we consider the Gerber-Shiu function in the Markov modulation risk model.By the method of probability we obtained integral-differential equations of ??(u;i)and?d(u;i)in the Markov modulation model perturbed by diffusion,and then the Laplace transform of ??(u;i)and ?d(u;i)is calculated by the derivative function of the integral-differential equation.At last,when the claim amount is exponential distribution the expression of ??(u;i)and ?d(u;i)is calculated.