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

Abstract

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 with H∈(1/2,1), 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=12. We apply a linearized generalized Newton method to construct an iterative sequence for the value function and provide a numerical convergence analysis via the contraction mapping theorem. Using real GOOGL data, we obtain a model-implied mean optimal allocation of π¯*=68.35% for H=0.62, close to the classical 69.76%. A comprehensive sensitivity analysis identifies volatility σ and jump intensity λ as the dominant drivers of the optimal allocation, with absolute effects of 9.44% and 6.47%, respectively, while the Hurst parameter H, the jump threshold τ, and mean return α have smaller effects. The proposed framework provides a dynamic optimal investment ratio π*(t) that adjusts to market memory, offering a model-based strategy for portfolio management under both jump and long-memory risks. Empirical validation through backtesting is needed to establish its practical superiority.