Research On The Optimal Investment And Reinsurance Problem Based On Correlation

With the development of insurance industry,the competition of insurance industry is also increasing.In order to survive and develop in the fierce insurance market,the optimal reinsurance and investment problem for an insurer has attracted more and more attention.Reinsurance and investment are two important ways for an insurer to control risks and increase wealth.Specifically,the insurers usually purchase reinsurance to cede part of their claim losses to the reinsurer and pay certain reinsurance premium to the reinsurer.At the same time,the insurer can increase their profit by investing in the financial market.In addition,the insurer should not only consider the long-term income and short-term in-come from investment,but also consider the claim risk from the policyholder.This means that the correlation between the insurance market and the financial market has to be con-cerned.This kind of correlation may be due to an extreme event that has the common impact on the financial and insurance markets,such as natural disasters,financial crises,serious epidemic diseases and so on.Therefore,the optimal reinsurance and investment problem based on the correlation between the insurance market and the financial market has vitally theoretical and practical significance.This paper studies the insurer’s opti-mal reinsurance and investment strategies based on the correlation between the financial market and the insurance market from the following perspectives.The first part of this thesis studies the optimal reinsurance and investment problem with default risk and correlation under the constant elasticity variance（CEV）model.The insurer’s claim process is described by a Brownian motion with drift.Suppose that the insurer can purchase proportional reinsurance and invest in a risk-free asset,a stock and a defaultable bond.The stock’s price process follows the CEV model.In addition,we consider the correlation between the risk model and the price of the stock.For the objec-tive of maximizing the expectation utility of the terminal wealth and the special elastility factor,we obtain the optimal reinsurance and investment strategies and the corresponding value function explicitly for the pre-default case and the post-default case by using the stochastic control method.The second part of this thesis considers the asymptotic solution of the optimal rein-surance and investment problem based on the correlation under the CEV model.The insurer’s claim process follows a Brownian motion with drift.The insurer is allowed to purchase proportional reinsurance and invest in a financial market.The financial market consists of a risk-free asset and a risky asset whose price process follows the CEV model.In addition,the correlation between the risk model and the risky asset’s price is taken into account.By applying the dynamic programming principle,we establish the Hamilton-Jacobi-Bellman（HJB）equation for the objective of the terminal wealth’s expected utility maximization.Inspired by Park（2011）^[1],we express the solution of partial differential equation derived from HJB equation in the form of asymptotic expansion,and then obtain the optimal reinsurance and investment strategies for an insurer by using the perturbation theory.In the third part of this thesis,we study the optimal reinsurance and investment prob-lem with multiple risky assets and correlation under the Ornstein-Uhlenbeck（O-U）model.As in the previous two parts,the insurer’s claim process follows the diffusion approxima-tion model.The insurer is allowed to invest in a risk-free asset and multiple risky assets,where the instantaneous return rate of each risky asset is described by the O-U process.In addition,we take the correlation between the risk model and each risky asset’s price into account.First,we study the optimal investment problem aiming at maximizing the ex-pectation utility of the terminal wealth.Secondly,we assume that the insurer can buy the proportional reinsurance and invest in the financial market.Then we consider the optimal reinsurance and investment problem for the insurer under the same objective.By using the dynamic programming principle,we derive the optimal reinsurance and investment strategies and the corresponding value function.The fourth part of this thesis investigates the optimal investment problem with multi-ple risky assets based on the correlation under the CEV model for an insurer.The insurer’s claim process is also described by a Brownian motion with drift.The insurance premium is calculated according to the mean-variance principle.We assume that the insurer can purchase proportional reinsurance and invest in a risky-free asset and multiple risky as-sets,where the price process of each risky asset follows the CEV model.In addition,the correlation between the risk model and each risky asset’s price is considered.The objective of the insurer is to maximize the expected utility of the terminal wealth.We first establish the general framework of the optimization problem.Then we propose a new expression of the value function under the exponential utility,which is different from the previous literature,and derive the insurer’s optimal investment strategy explicitly.Be-sides,we consider some special cases of our model,that is,the optimal problem with a risky asset and with two risky assets,and obtain the optimal investment strategy for an insurer.This thesis aims to establish a more practical mathematical model and studies the impact of insurance market and financial market on the insurer’s optimal reinsurance and investment strategies.This can provide some theoretical basis and reference value for an insurer in reinsurance purchase and insurance investment.In order to improve the maneuverability of the study and visually display the model results,we provide some numerical examples at the end of each study to illustrate the sensitivity of the insurer’s optimal strategy to the model parameters.