<p>This study delves into the captivating interplay of space-time fractional dynamics within the Hamilton-Jacobi-Bellman model, a pivotal framework in stochastic optimal control with far-reaching implications for finance, physics, and engineering. By harnessing an innovative computational approach, we unravel the complexities of fractional memory and spatial diffusion, offering fresh insights into their influence on system behavior. Our findings, supported by rigorous theoretical analysis and compelling numerical results, highlight the profound effects of fractional orders on wealth evolution under transaction costs, unveiling intricate patterns that shape decision-making processes. With applications ranging from portfolio optimization and wealth management to turbulence modeling, this work bridges cutting-edge mathematics with real-world challenges, opening new avenues for tackling stochastic control problems and inspiring future explorations in this dynamic field.</p>