Jump-diffusion optimization: An iterative solution to the HJB equation for investment value

Paziresh, Mehran; Ivaz, Karim

doi:10.22034/cmde.2025.68581.3328

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	Jump-diffusion optimization: An iterative solution to the HJB equation for investment value
Computational Methods for Differential Equations
مقاله 11، دوره 14، شماره 2، تیر 2026، صفحه 661-677 اصل مقاله (879.99 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2025.68581.3328
نویسندگان
Mehran Paziresh؛ Karim Ivaz^*
Faculty of Mathematics, Statistics, and Computer Sciences, University of Tabriz, Tabriz, Iran.
چکیده
This paper focuses on optimizing the investment value function by incorporating jump risk using the Merton Jump Diffusion (MJD) model. Our main goal is to determine the optimal dynamic asset allocation strategy to maximize expected utility. We derive the governing nonlinear Hamilton-Jacobi-Bellman (HJB) equation and employ a linearized generalized Newton method, which generates an iterative sequence for the optimal control. The theoretical convergence of this sequence was rigorously established using the Contraction Mapping Theorem, confirming the method's strong stability and reliability. Applying the model to real Google stock data, which exhibit significant jump risks, we derived an optimal investment ratio ($\mathbf{\pi^*}$) that suggests a notably aggressive allocation to the risky asset. This optimal strategy provides a direct, actionable benchmark for investors. Crucially, the derived dynamic control law functions as a powerful tool for investment management firms, enabling them to proactively adjust capital allocation strategies in response to potential future jump risk scenarios.
کلیدواژه‌ها
Investment value function؛ Jump risk؛ Merton Jump-diffusion model؛ Hamilton-Jacobi-Bellman equation؛ Contraction mapping؛ Frechet derivative

مراجع
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Jump-diffusion optimization: An iterative solution to the HJB equation for investment value