Optimal Reinsurance And Investment To Minimize The Probabilities Of Drawdown And Absolute Ruin

Risk control and management has long been an important topic for insurance and financial firms.On the one hand,they need to reduce risk exposure by purchasing reinsurance.On the other hand,they want to get more profit by investing their surplus into the risky and risky-free asset.Therefore,the research on the optimal reinsurance and investment problems for insurers is becoming a focus in insurance and actuarial literature,with high academic values and broad applications prospects.In this dissertation,we mainly consider two important risk-measure criteria,minimizing the probability of drawdown and minimizing the probability of absolute ruin.With drawdown,the decision-makers want to adopt strategies which minimize the probability that the value of the surplus process drops below some fixed proportion,say ? ? [0,1),of its maximum value to date.It is not difficult to verify that when? > 0,the criterion of minimizing the probability of drawdown is a more conservative and general criterion than the one of minimizing the probability of traditional ruin and it includes minimizing the probability of ruin as a special case when ? = 0.Absolute ruin probability is also an important risk measure and has been considered in some research works.Unlike the study of traditional ruin probability with ruin level 0,the study of absolute ruin probability allows us to investigate the behavior of a company when it is in deficit.Therefore,compared with traditional ruin probability minimization problem,the absolute ruin probability minimization problem is a kind of more radical and general risk measure.In some papers,absolute ruin occurs when the surplus rate level drops below a critical level(the premiums received are not sufficient to make the interest payments on the debt).Some other papers defined the absolute ruin as an event that lim inf of the surplus process is negative infinity.Since the domains and boundary conditions for drawdown and absolute ruin problems are very different from the traditional ruin,the optimization problems become much more complicated and practical.Based on existing literatures,by the techniques of stochastic control theory and the corresponding the Hamilton-Jacobi-Bellman(HJB)equation,this thesis carries out the following aspects' research on the optimal reinsurance and investment strategy for an insurer under the criteria of minimizing the probability of drawdown and minimizing the probability of absolute ruin:Since insurance businesses are usually dependent in some way,it is reasonable to consider some dependence structures among several classes of insurance business.Under the expected value principle,Chapter 2 studies the optimal proportional reinsurance strategy in a risk model with two dependent classes of insurance business,where the two claim number processes are correlated through a common shock component,and the criterion is to minimize the probability of drawdown.Then we generalize the dependence structure to the thinning dependence structure.The optimal reinsurance strategy and the minimum probability of drawdown are derived not only for the expected value principle but also for the variance premium principle.When the surplus is relatively low,the insurer prefers to pay more attention to reducing the risk;but when the surplus becomes relatively high,the insurer may be more interested in reaching a revenue target as quickly as possible.Therefore,assuming that the insurer takes both investment and reinsurance into consideration and the price process of risky asset is correlated to the claim process,Chapter 3 considers the objectives of survival and growth in two complementary regions.The optimal results for both aspects of the problems therefore complement the results in Chapter 2.Much of the literature considers problems under which the insurance company is constrained to buy either pure quota-share reinsurance,pure excess-of-loss reinsurance,or a combination of the two.Chapter 4 derives the optimal reinsurance strategy to minimize the probability of drawdown without restricting the form of the reinsurance and the reinsurance premium is computed according to the mean-variance premium principle,a combination of the expected value and variance premium principle.We observe that the optimal reinsurance strategy is in the form of pure excess-of-loss reinsurance strategy under the expected principle;and under the variance premium principle,the optimal reinsurance strategy is in the form of the pure quota-share reinsurance.It turns out that this conclusion can also be applied to the risk model with two dependent classes of insurance business.Then based on the general risk model in Chapter 4,we turn our attention to explore the optimal reinsurance and investment strategies under the objective of minimizing the probability of absolute ruin in Chapter 5.We show that when the absolute ruin is defined as an event that lim inf of the surplus process is negative infinity,the value function is no longer convex as the one in the traditional ruin problem,but is S-shaped with a unique point of inflection.We further incorporate model uncertainty into an insurer's controlled surplus and aim to minimize the robust value involving the probability of absolute ruin and a penalization of model ambiguity.We find that the optimal robust reinsurance strategy with ambiguity penalization is independent of the penalty parameter,and the robust value function is still S-shaped.However,due to the fact that the value function depends on the penalty parameter greatly,the resulting position of the inflection point is different from the one without model ambiguity.This thesis is devoted to making the problem more practical,and trying to give the explicit expressions of the optimal reinsurance and investment strategies.Some numerical examples are presented to illustrate our results.