Optimal Insurance Pricing and Demands in a Stochastic Stackelberg Differential Game

Abstract

This paper investigates portfolio selection and risk control between individuals and an insurer in a frame of stochastic Stackelberg differential game. In the Stackelberg game, the insurer plays as the leader who determines the price of insurance, and individuals play as the followers deciding how much insurance they should buy. On one hand, we suppose that there are $n$ homogeneous individuals in the market who can purchase insurance from one insurer. Each individual aims to achieve optimal consumption, insurance, and investment policies while considering premium prices and the performance of financial market. On the other hand, the insurer can adjust premium prices and investment policies in response to the individuals' insurance demands and the performance of financial market. Both individuals and the insurer aim to maximize their expectation of utilities. By employing the dynamic programming principle, we obtain closed-form solutions for optimal policies and value functions. The results show that the correlation between financial market and insurance market has a significant impact on the optimal strategy. A strong negative correlation would lead to an inefficient insurance market, but full insurance demand occurs for strong positive correlated markets. Most interestingly, our result indicates that insurance demand can effectively increase individuals consumption level and investment level on risk assets for positive correlated markets. Finally, we also find the equilibrium insurance price decreases and the equilibrium insurance demand increases with respect to the size of insurance market for positive correlated markets.