Optimal portfolio under fractional stochastic environment

Jean‐Pierre Fouque and Ruimeng Hu

Mathematical Finance, 2019, vol. 29, issue 3, 697-734

Abstract: Rough stochastic volatility models have attracted a lot of attention recently, in particular for the linear option pricing problem. In this paper, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non‐Markovian) fractional stochastic environment (for all values of the Hurst index H∈(0,1)). We rigorously establish a first‐order approximation of the optimal value, when the return and volatility of the underlying asset are functions of a stationary slowly varying fractional Ornstein–Uhlenbeck process. We prove that this approximation can be also generated by a fixed zeroth‐ order trading strategy providing an explicit strategy which is asymptotically optimal in all admissible controls. Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this fixed strategy in a specific family of admissible strategies.

Date: 2019
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