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.
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Paziresh, M. and Ivaz, K. (2026). Jump-diffusion optimization: An iterative solution to the HJB equation for investment value. Computational Methods for Differential Equations, 14(2), 661-677. doi: 10.22034/cmde.2025.68581.3328
MLA
Paziresh, M. , and Ivaz, K. . "Jump-diffusion optimization: An iterative solution to the HJB equation for investment value", Computational Methods for Differential Equations, 14, 2, 2026, 661-677. doi: 10.22034/cmde.2025.68581.3328
HARVARD
Paziresh, M., Ivaz, K. (2026). 'Jump-diffusion optimization: An iterative solution to the HJB equation for investment value', Computational Methods for Differential Equations, 14(2), pp. 661-677. doi: 10.22034/cmde.2025.68581.3328
CHICAGO
M. Paziresh and K. Ivaz, "Jump-diffusion optimization: An iterative solution to the HJB equation for investment value," Computational Methods for Differential Equations, 14 2 (2026): 661-677, doi: 10.22034/cmde.2025.68581.3328
VANCOUVER
Paziresh, M., Ivaz, K. Jump-diffusion optimization: An iterative solution to the HJB equation for investment value. Computational Methods for Differential Equations, 2026; 14(2): 661-677. doi: 10.22034/cmde.2025.68581.3328
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