Stochastic analysis of an economic growth model incorporating Itô–Lévy driven investment, optimal control and numerical simulation

In this paper, we develop and analyze a stochastic model of economic growth to determine the optimal production trajectory under uncertainty. The source of uncertainty lies in the capital dynamics, modeled through a stochastic differential equation driven by Lévy processes. By applying Hamilton–Jacobi–Bellman (HJB) optimization techniques adapted to Lévy processes, we derive an optimal investment–consumption policy. To address the complexity of the resulting nonlinear system, we extend the New Local Linearization method to accommodate jump–diffusion processes. Compared to the case without jumps, our findings demonstrate that incorporating jump components into the investment model amplifies the role of jump-induced volatility and leads to a higher elasticity of capital. We conclude that Itô–Lévy formulations not only allow for closed-form solutions to Gross Domestic Product (GDP) dynamics but also provide a robust and reproducible framework, offering valuable insights and directions for future research in economic and stochastic modeling.