On Utility Maximization under Multivariate Fake Stationary Affine Volterra Models

TL;DR

This paper addresses Merton's portfolio optimization in a complex multivariate Volterra environment, employing Riccati BSDEs to derive optimal strategies despite non-Markovian challenges.

Contribution

It introduces a novel approach using Riccati BSDEs for portfolio optimization under multivariate fake stationary Volterra models, overcoming non-Markovian difficulties.

Findings

Optimal strategies are expressed in semi-closed form via Riccati equations.

Numerical results show the impact of stationary rough volatilities on strategies.

The approach extends classical methods to non-Markovian stochastic environments.

Abstract

This paper is concerned with Merton's portfolio optimization problem in a Volterra stochastic environment described by a multivariate fake stationary Volterra--Heston model. Due to the non-Markovianity and non-semimartingality of the underlying processes, the classical stochastic control approach cannot be directly applied in this setting. Instead, the problem is tackled using a stochastic factor solution to a Riccati backward stochastic differential equation (BSDE). Our approach is inspired by the martingale optimality principle combined with a suitable verification argument. The resulting optimal strategies for Merton's problems are derived in semi-closed form depending on the solutions to time-dependent multivariate Riccati-Volterra equations, while the optimal value is expressed using the solution to this original Riccati BSDE. Numerical results on a two dimensional fake stationary rough Heston model illustrate the impact of stationary rough volatilities on the optimal Merton strategies.