We show how to solve Merton optimal investment stochastic control problem for Hawkes-based models in finance and insurance (Propositions 1 and 2), i.e., for a wealth portfolio $X\left(t\right)$ consisting of a bond and a stock price described by general compound Hawkes process (GCHP), and for a capital $R\left(t\right)$ (risk process) of an insurance company with the amount of claims described by the risk model based on GCHP. The main approach in both cases is to use functional central limit theorem for the GCHP to approximate it with a diffusion process. Then we construct and solve Hamilton–Jacobi–Bellman (HJB) equation for the expected utility function. The novelty of the results consists of the new Hawkes-based models and in the new optimal investment results in finance and insurance for those models.

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基于霍克斯模型的金融和保险领域的默顿投资问题

我们展示了如何解决金融和保险中基于霍克斯模型的默顿最优投资随机控制问题（命题 1 和 2），即财富投资组合 $X\left(吨\right)$ 由一般复合霍克斯过程 (GCHP) 描述的债券和股票价格组成，对于资本 $\mathrm{电阻}\left(吨\right)$（风险过程）保险公司的索赔金额由基于 GCHP 的风险模型描述。这两种情况下的主要方法是使用 GCHP 的泛函中心极限定理来近似它的扩散过程。然后我们构造并求解 Hamilton-Jacobi-Bellman (HJB) 方程以获得预期效用函数。结果的新颖性包括新的基于霍克斯的模型以及这些模型在金融和保险方面的新最佳投资结果。