AI 中文总结
研究随机利率和跳跃驱动负债下保险公司最优盈余管理问题，利用随机控制技术推导HJB方程，采用归一化盈余投影方法简化问题，得出最优投资策略含特定成分，通过数值实验揭示其与相关因素的关系及联合建模的重要性。
AI 中文摘要
本文研究了在具有随机利率和跳跃驱动负债的金融市场中运营的保险公司的最优盈余管理问题。保险公司在面临由复合泊松过程建模且索赔规模呈指数分布的保险索赔时，在风险股票和无风险零息债券之间动态分配其盈余。短期利率遵循Cox-Ingersoll-Ross（CIR）过程。保险公司最大化终端盈余的预期指数效用。利用随机控制技术推导相关的Hamilton-Jacobi-Bellman（HJB）方程。采用归一化盈余投影方法将三维问题简化为非线性偏微分方程组并进行数值验证。最优投资策略包括近视需求成分和利率套期保值成分。数值实验表明最优策略和盈余分布如何依赖于利率波动率、索赔强度和风险厌恶。结果强调了在为保险公司设计最优投资策略时联合建模随机利率和保险负债风险的重要性。
英文摘要
This paper investigates the optimal surplus management problem of an insurance company operating in a financial market with stochastic interest rates and jump-driven liabilities. The insurer dynamically allocates its surplus between a risky stock and a risk-free zero-coupon bond while facing insurance claims modeled by a compound Poisson process with exponentially distributed claim sizes. The short term interest rate follows a Cox-Ingersoll-Ross (CIR) process, which captures mean-reverting dynamics commonly observed in term structure models. The insurer maximizes the expected exponential utility of terminal surplus. Using stochastic control techniques, we derive the associated Hamilton-Jacobi-Bellman (HJB) equation. Although the exponential utility structure suggests an exponential affine representation, the interaction between the interest rate hedge and the surplus state generates quadratic surplus terms in the HJB equation. To obtain a tractable formulation, we adopt a normalized surplus projection method, which provides an approximate reduction of the full three-dimensional problem to a nonlinear system of partial differential equations (which is subsequently numerically validated). The optimal investment policy admits an economically meaningful decomposition consisting of a myopic demand component and an interest rate hedging component. Numerical experiments illustrate how the optimal strategy and the surplus distribution depend on interest rate volatility, claim intensity, and risk aversion. The results highlight the importance of jointly modeling stochastic interest rates and insurance liability risk when designing optimal investment policies for insurance companies.
