基于新巴塞尔协议监管下保险人的均值-方差最优投资-再保险问题

Optimal Mean-Variance Investment-Reinsurance Problem with Constrained Controls by the New Basel Regulations for an Insurer

本文研究了均值-方差优化准则下,保险人的最优投资和最优再保险问题.我们用一个复合泊松过程模型来拟合保险人的风险过程,保险人可以投资无风险资产和价格服从跳跃-扩散过程的风险资产.此外保险人还可以购买新的业务(如再保险).本文的限制条件为投资和再保险策略均非负,即不允许卖空风险资产,且再保险的比例系数非负.除此之外,本文还引入了新巴塞尔协议对风险资产进行监管,使用随机二次线性(linear-quadratic,LQ)控制理论推导出最优值和最优策略.对应的哈密顿-雅克比-贝尔曼(Hamilton-Jacobi-Bellman,HJB)方程不再有古典解.在粘性解的框架下,我们给出了新的验证定理,并得到有效策略(最优投资策略和最优再保险策略)的显式解和有效前沿.

We study the optimal investment and optimal reinsurance problem for an insurer under the criterion of mean-variance.The insurer’s risk process is modeled by a compound Poisson process and the insurer can invest in a risk-free asset and a risky asset whose price follows a jump-diffusion process.In addition,the insurer can purchase new business(such as reinsurance).The controls(investment and reinsurance strategies) are constrained to take nonnegative values due to nonnegative new business and no-shorting constraint of the risky asset.We control the risk by the new Basel regulation and use the stochastic linear-quadratic(LQ) control theory to derive the optimal value and the optimal strategy.The corresponding Hamilton-Jacobi-Bellman(HJB) equation no longer has a classical solution.With the framework of viscosity solution,we give a new verification theorem,and then the efficient strategy(optimal investment strategy and optimal reinsurance strategy) and the efficient frontier are derived explicitly.