Optimal Investment And Risk Control Policies For An Insurer In An Incomplete Market

The optimal investment and risk control for an insurer exerted a crucial part in mathematical finance,which has attracted extensive attention of the researchers at home and abroad.In the past,most researchers considered the problem of optimal investment via establishing and solving the corresponding HJB equation.In this paper,we investigate the optimal investment and risk control problem for an insurer,who aims to maximize the expected utility of terminal wealth in an incomplete market by the martingale approach.This paper briefly summarizes the historical background and influence of the risk theory,mainly introduces several kinds of classical risk model and gives the basic knowledge and some theorems of stochastic analysis which relative to the martingale approach.Based on literatures,according to the martingale,we investigate the opti-mal investment and risk control policies with multiple risky assets whose price processes are geometric Brownian motions and a risk-free asset in an incom-plete market.The risk process of per policy is described by a pure-jump Levy process and the insurer can regulate the risk by controlling the number of in-surance policy.However,in an incomplete market,the martingale method will be problematic.Thus we will reduce the dimension of the Brownian motion to transform the incomplete market into a complete one.The optimal invest-ment and risk control problems under mean-variance criterion is studied by the martingale approach.Furthermore,numerical calculations are presented to illustrate the effects of various parameters of the model on the optimal invest-ment and risk control and the efficient frontier.Finally,aming to maximize the expected utility of terminal wealth,we obtain the closed-form solutions to the problem of optimal investment and risk control under the constant risk aversion utility via the martingale approach.