Mean-variance investment and risk control strategies for a dynamic contagion process with diffusion

This paper explored an investment and risk control issue within a contagious financial market, specifically focusing on a mean-variance (MV) framework for an insurer. The market's risky assets were depicted via a jump-diffusion model, featuring jumps due to a multivariate dynamic contagion process with diffusion (DCPD). The process enveloped several popular processes, including the Hawkes process with exponentially decaying intensity, the Cox process with Poisson shot-noise intensity, and the Cox process with Cox-Ingersoll-Ross (CIR) intensity. The model distinguished between externally excited jumps, indicative of exogenous influences, modeled by the Cox process, and internally excited jumps, representing endogenous factors captured by the Hawkes process. Given an expected terminal wealth, the insurer seeked to minimize the variance of terminal wealth by adjusting the issuance volume of policies and investing the surplus in the financial market. In order to address this MV problem, we employed a suite of mathematical techniques, including the stochastic maximum principle (SMP), backward stochastic differential equations (BSDEs), and linear-quadratic (LQ) control techniques. These methodologies facilitated the derivation of both the efficient strategy and the efficient frontier. The presentation of the results in a semi-closed form was governed by a nonlocal partial differential equation (PDE). For empirical validation and demonstration of our methodology's efficacy, we provided a series of numerical examples.