ABSTRACT

We consider the problem of to which extent a diffusion process serves as a valid approximation of the classical Cramér-Lundberg (CL) risk process for a Stackelberg differential game between a buyer and a seller of insurance. We show that the equilibrium for the diffusion approximation equals the limit of the equilibrium for the scaled CL process, and it is nearly optimal for the pre-limit problem. Specifically, if the loss process follows a CL risk process and ambiguity is measured via entropic divergence, then the Stackelberg equilibrium of the diffusion approximation with squared-error divergence approximates the equilibrium for the former model to order $\mathcal{O}(\frac{1}{\sqrt{n}})$, in which we scale the CL model via n, as in Cohen and Young [(2020). Rate of convergence of the probability of ruin in the Cramér-Lundberg model to its diffusion approximation. Insurance: Mathematics and Economics 93: 333–340].

摘要

我们考虑扩散过程在多大程度上可以有效近似于保险买卖双方之间的 Stackelberg 差异博弈的经典 Cramér-Lundberg (CL) 风险过程的问题。我们表明，扩散近似的平衡等于缩放 CL 过程的平衡极限，并且对于预极限问题几乎是最优的。具体来说，如果损失过程遵循 CL 风险过程，并且通过熵散度测量模糊度，则具有平方误差散度的扩散近似的 Stackelberg 平衡近似于前一个模型的平衡$\mathcal{○}(\frac{1}{\sqrt{n}})$，其中我们通过n缩放 CL 模型，如 Cohen 和 Young [(2020). Cramer-Lundberg 模型中破产概率与其扩散近似值的收敛速度。保险：数学和经济学93：333-340]。