Viscosity Solutions of Stochastic HJB Equations for Utility Maximization in Fractional SABR Models
Abel ZONGO *
Departement de Mathematiques, Universite Joseph KI-ZERBO, 03 BP 7021, Ouagadougou, Burkina Faso.
Frederic NIKIEMA
Departement de Mathematiques, Ecole Normale Superieure, 01 BP 1757, Ouagadougou 01, Burkina Faso.
*Author to whom correspondence should be addressed.
Abstract
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.
Keywords: Optimal stochastic control, fractional stochastic Hamilton-Jacobi-Bellman equation, viscosity solution