关键词: Heston’s SV模型, 确定性利率函数, 动态规划原理, 稳键的再保险-投资, 模糊不确定

Abstract: The insurance company is an enterprise operating and managing risks. In order to avoid huge losses or even bankruptcy caused by excessive risks in the future operation of insurance companies, on the one hand, insurance companies will use reinsurance to share its risks. On the other hand, insurance companies will make reasonable and effective investment strategies to increase the stability of their own operations. For the past decades, many experts and scholars have studied the optimal reinsurance and investment problems for insurance companies. In general, the optimal reinsurance-investment strategies of insurance companies are considered under three kinds of objective functions. The first kind aims to minimize the ruin probability of insurance company. The second is to maximize the expected exponential utility of terminal wealth of the insurance company. The third type is the mean-variance criterion.
For the past years, many scholars have found that stochastic volatility is an important feature of stock price models, as it can better explain the volatility smile, the thick-tailed nature of return distribution, and other features of stock prices. In general, some papers consider the optimal reinsurance and investment strategies of an insurer, where the prices of risk assets are described by GBM (Geometric Brownian Motion) model, O-U (Ornstein-Uhlenbeck process) model and CEV (Constant Elasticity of Variance) model. Compared with the Heston’s SV (Heston’s Stochastic Volatility) model, none of these models contain full-fledged stochastic volatility assumptions. Moreover, on the one hand, insurance companies have uncertainty in their risk selection preferences when making optimal portfolios. On the other hand, it is difficult to accurately estimate the rate of return of risky assets and the expected surpluses in portfolio management. Therefore, relative to theAmbiguity-Neutral Insurer (ANI), Ambiguity-Averse Insurer (AAI) will look for a way to deal with this uncertainty by considering some alternative models close to the estimated model to arrive at a robust optimal reinsurance and investment strategies in the event of model uncertainty. And this systematic approach has been successfully implemented in portfolio selection and asset pricing, and already has some applications in insurance.
In order to better fit the investment situation of insurers in real financial markets, we consider the optimal reinsurance-investment issues of AAI based on Heston’s SV model under variable interest rate. The paper assumes that the AAI, after taking the proportional reinsurance into account, allocates its surplus between the risk-free asset and the risky asset, where the price process of the risk-free asset follows the differential equation derived by deterministic interest rate function and the price process of the risky asset follows the Heston’s stochastic volatility model, respectively. The goal of our optimal problem is to minimize the maximal expected exponential utility of terminal wealth on a set of absolutely continuous probability measures. Firstly, we describe the alternative model by the probability measure that is equivalent to the probability measure of the reference model in the case of model uncertainty. By Girsanov transformation, the equivalent wealth process of an insurer under the alternative model is derived. Then, we establish the corresponding HJB (Hamilton-Jacob-Bellman)equation by applying the stochastic dynamic programming approach, which measures the ambiguity of model uncertainty with different preference parameters with state dependence. And explicitly expressions of the optimal robust reinsurance and investment strategies are derived by solving HJB equation with the CARA (Constant Absolute Risk Aversion) utility function. Finally, the impact of each parameter on the optimal reinsurance and investment is obtained through numerical simulations and the corresponding economic analysis is presented.
The innovation of this paper is twofold. Under variable interest rates, we establish a robust optimal reinsurance and investment model framework for AAI based on Heston’s SV model. In this framework, we not only consider variable interest rates, but also consider the influence of different state preference parameters describing model uncertainty on the optimal reinsurance and investment strategies for an insurer. The results show that the expression of the optimal reinsurance and investment strategies are more accurate than that of the model using the same preference parameter. In addition, the Heston’s SV model that we think about at variable interest rates is more consistent with the real financial environment. Through the numerical simulations, the results and corresponding economic analysis can provide some theoretical guidance for insurers to carry out reinsurance and investment in the financial markets. Furthermore, options can be considered in the investment process of risky assets to enrich the types of investment. Taxes and transaction costs can also be added to make the model more suitable for the actual financial environment. Moreover, we can also consider the optimal reinsurance and investment strategy problem under Heston’s SV model in the case of inflation. This is also the problem that we will consider in the future.

Key words: Heston’s SV model, deterministic interest rate function, dynamic programming approach, robust reinsurance and investment, ambiguity