Optimal investment and premium control in a nonlinear diffusion model
Optimal investment and premium control in a nonlinear diffusion model
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DOI：
10.1007/s10255-017-0709-7
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发表时间：
2017-10
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影响因子：
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通讯作者：
M. Zhou;K. Yuen;C. Yin
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文献类型：
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作者：
M. Zhou;K. Yuen;C. Yin
本文考虑了一类非齐次复合Poisson过程的扩散近似下的最优投资与保费控制问题。在非线性扩散模型中，假设存在一个未知的单调函数来描述保费的安全负荷与时变索赔到达率之间的关系。因此，除了投资控制，保险费率可以作为一个控制变量的优化问题。具体地说，该问题被研究在两种情况下：（i）最大化的预期效用的终端财富，（ii）最小化的破产概率分别。在这两种情况下，价值函数的一些性质的推导，最优策略和价值函数的封闭形式的表达式。结果表明，最优投资策略和最优保费控制策略是相互依赖的。最有趣的是，作为一个例子，我们表明，非线性扩散模型减少到一个二次漂移系数的扩散模型时，与保险费率和索赔到达率的函数采取一种特殊的形式。实例表明，所研究的模型代表了一类非线性随机控制风险模型。
This paper considers the optimal investment and premium control problem in a diffusion approximation to a non-homogeneous compound Poisson process. In the nonlinear diffusion model, it is assumed that there is an unspecified monotone function describing the relationship between the safety loading of premium and the time-varying claim arrival rate. Hence, in addition to the investment control, the premium rate can be served as a control variable in the optimization problem. Specifically, the problem is investigated in two cases: (i) maximizing the expected utility of terminal wealth, and (ii) minimizing the probability of ruin respectively. In both cases, some properties of the value functions are derived, and closed-form expressions for the optimal policies and the value functions are obtained. The results show that the optimal investment policy and the optimal premium control policy are dependent on each other. Most interestingly, as an example, we show that the nonlinear diffusion model reduces to a diffusion model with a quadratic drift coefficient when the function associated with the premium rate and the claim arrival rate takes a special form. This example shows that the model of study represents a class of nonlinear stochastic control risk model.