Viscosity Solutions of Stochastic HJB Equations for Utility Maximization in Fractional SABR Models

This work addresses the fundamental problem of portfolio optimization in quantitative finance, which aims to maximize returns for a given level of risk. Building on previous research, this study analyzes a portfolio composed of a risky asset and a less-risky asset. The main contribution is to solve the optimization problem by using the Hamilton-Jacobi-Bellman (HJB) equation to approximate the solution of a fractional stochastic partial differential equation (FSPDE), which models the optimal investment strategy. Our main theoretical results establish the existence and uniqueness of a viscosity solution and provide a semi-analytical expression for the optimal investment strategy. From a practical standpoint, this framework provides portfolio managers with a more realistic tool for asset allocation in financial markets exhibiting long-memory effects.