Optimal Investment Control Under Jump-Fractional Dynamics: A Wick–Itô Approach

This paper extends the optimal investment control framework by incorporating fractional Brownian motion to capture long-range dependence and memory effects in asset prices. Replacing the standard Brownian component with a fractional Brownian motion governed by the Hurst parameter H, we employ the Wick–Itô calculus to derive the associated Hamilton–Jacobi–Bellman (HJB) equation. The resulting nonlinear PDE contains a time-dependent diffusion coefficient that reduces to the classical model when H = ½. We apply a linearized generalized Newton method to construct an iterative sequence for the value function, and prove its convergence via the contraction mapping theorem. The proposed framework provides a dynamic optimal investment ratio π*(t) that adjusts to market memory, offering a more realistic strategy for portfolio management under both jump and long-memory risks.