Optimal portfolio selection in jump-uncertain stochastic markets via maximum principle and dynamic programming

Clift Kudzai Hlahla^1*Eriyoti Chikodza²Cathrine Kazunga¹Mangwiro Magodora¹

• ¹Bindura University of Science Education, Bindura, Zimbabwe
• ²University of Botswana, Gaborone, Botswana

This paper develops a unified framework for optimal portfolio selection in jump–uncertain stochastic markets, contributing both theoretical foundations and computational insights. We establish the existence and uniqueness of solutions to jump–uncertain stochastic differential equations, extending earlier results in uncertain–stochastic and Liu–uncertain settings without jumps, and provide a rigorous proof of the principle of optimality, thereby reinforcing the link between dynamic programming and the maximum principle under both continuous and discontinuous uncertainty. Applying this framework to a financial market with jump uncertainty, we demonstrate that under constant relative risk aversion (CRRA) utility, the optimal portfolio rule preserves the constant–proportion property, remaining independent of wealth. Numerical analysis further reveals consistent comparative statics: the optimal fraction ρ allocated to the risk–free asset rises with Brownian volatility σ1 and jump intensity λ, reflecting precautionary behavior under uncertainty, while it declines with the expected risky return µ and the risk–aversion parameter κ, indicating greater exposure to risk when returns are higher or investors are less risk–averse. Taken together, these results confirm the robustness, tractability, and economic relevance of the framework, aligning with classical findings in jump–diffusion models and offering implementable strategies for decision making in financial markets subject to both continuous and jump risks.