Legendre Transform Dual Asymptotic Solution for Insurers Under the Heston Local‐Stochastic Volatility Model: A Comparison of Variance Premium and Expected Value Principles

This study examines optimal investment and reinsurance strategies for two competing insurers who are concerned with their relative performance. Each insurer can purchase reinsurance and invest in a financial market consisting of one risk‐free asset and one risky asset, with the risky asset’s price modeled using the Heston local‐stochastic volatility (HLSV) model, which combines the characteristics of both the CEV and Heston models. When optimizing strategies under an exponential utility function, an analytical solution is not attainable due to the complex nonlinearity of the resulting partial differential equation. To address this, we employ a dual method, Legendre transformation, and an asymptotic expansion technique to obtain an approximate solution considering only the slow‐varying volatility factor. The analysis is conducted under two premium calculation frameworks: the variance premium principle in the first part and the expected value principle in the second part. Finally, we complement the theoretical findings with numerical studies and provide economic interpretations for the optimal reinsurance strategies derived under both principles.