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

Xiuxian Chen; Zhongyang Sun; Dan Zhu; Xiuxian Chen; Zhongyang Sun; Dan Zhu

doi:10.3934/math.20241580

AIMS Mathematics

2024, Volume 9, Issue 11: 33062-33086. doi: 10.3934/math.20241580

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Research article

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

School of Statistics and Data Science, Qufu Normal University, Qufu, Shandong 273165, China

Received: 18 September 2024 Revised: 06 November 2024 Accepted: 15 November 2024 Published: 21 November 2024
MSC : 91G05, 91G10, 93E20

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.
Citation: Xiuxian Chen, Zhongyang Sun, Dan Zhu. Mean-variance investment and risk control strategies for a dynamic contagion process with diffusion[J]. AIMS Mathematics, 2024, 9(11): 33062-33086. doi: 10.3934/math.20241580

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Abstract

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.

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