Research On Optimal Investment,Consumption And Reinsurance Problems

As an important investment way with the roles of both insurance and savings,purchasing life-insurance is getting increasingly popular.Purchasing life-insurance has long been considered a protective tool to safeguard financial interests of dependents of policyholders in the event of their untimely deaths.The benefit paid upon death of the policyholder offers nominated dependents a source of living.Based on the policyholder's perspective,this article mainly investigates the optimal life-insurance investment and consumption problems in different cases.Moreover,as an important part of financial markets,insurers need to manage risk while providing insurance business for policyholders.On the one hand,insurers can transfer part of their risk to reinsurer by purchasing reinsurance.On the other hand,insurers can invest their surpluses in financial markets to achieve their management objectives.Therefore,it is important to find the optimal investment and reinsurance strategies for insurers.By using mean-variance theory,we also consider a time-consistent optimal investment and reinsurance problem for insurers.The content of this thesis is as follows:From the first perspective,we propose an optimal investment,consumption and life-insurance purchase problem for a wage earner.We assume that the price of the risky asset is governed by a continuous-time,finite state self-exciting threshold model.In this model,the state space of the price of the risky asset is partitioned by a set of thresholds and the parameters depend on the region which the current value of the price falls in.The wage earner's objective is to find the optimal investment,consumption and insurance strategies that maximize the expected discounted utilities.The optimal strategies for power utility function are derived by the martingale approach and the dynamic programming principle.Numerical examples are also provided to illustrate the effect of the thresholds.Regarding the second perspective,we present a technique to solve the problem where a couple aims to optimize their investment,consumption and lifeinsurance purchasing strategies,thereby maximizing their family objective until retirement.Assumed correlated lifetimes of the two wage earners are modeled by using both the copula and common-shock models.Subsequently,the explicit form solutions are obtained for determination of the optimal strategies in both the copula and a special case of the common-shock models.As observed,use of the copula model is more advantageous in its provision of closed-form strategies and ability to distinguish the impacts of mortality dependence.The optimization problem considered herein is investigated under a Markovian setting and solved using the Hamilton-Jacobi-Bellman equation.Numerical examples are also provided to illustrate the utility of the proposed optimization strategy.Finally,we also discuss a time-consistent investment and reinsurance problem for mean-variance insurers.We consider an insurer who manages her underlying risk by purchasing proportional reinsurance and making investment in a financial market consisting of a risk-free asset and multiple risky assets.The objective of the insurer is to identify an investment-reinsurance strategy that minimizes the mean-variance cost functional.We obtain a time-consistent equilibrium strategy and the corresponding efficient frontier in explicit form using two systems of backward stochastic differential equations.Furthermore,we apply our results to Vasi?cek stochastic interest rate model and Heston's stochastic volatility model.In both cases,we obtain a closed-form solution.