On the mean-variance problem through the lens of multivariate fake stationary affine Volterra dynamics

TL;DR

This paper addresses the mean-variance portfolio optimization in complex non-Markovian markets modeled by multivariate fake stationary affine Volterra processes, deriving explicit solutions via Riccati BSDEs.

Contribution

It introduces a novel approach using Riccati backward stochastic differential equations to solve the mean-variance problem in non-Markovian, non-semimartingale markets with explicit formulas.

Findings

Explicit closed-form expressions for optimal portfolios and efficient frontier.

Numerical experiments demonstrate the impact of rough volatility and stochastic correlations.

The method extends classical portfolio theory to complex Volterra-driven market models.

Abstract

We investigate the continuous-time Markowitz mean-variance portfolio selection problem within a multivariate class of fake stationary affine Volterra models. In this non-Markovian and non-semimartingale market framework with unbounded random coefficients, the classical stochastic control approach cannot be directly applied to the associated optimization task. Instead, the problem is tackled using a stochastic factor solution to a Riccati backward stochastic differential equation (BSDE). The optimal feedback control is characterized by means of this equation, whose explicit solutions is derived in terms of multi-dimensional Riccati-Volterra equations. Specifically, we obtain analytical closed-form expressions for the optimal portfolio policies as well as the mean-variance efficient frontier, both of which depend on the solution to the associated multivariate Riccati-Volterra system. To illustrate our results, numerical experiments based on a two dimensional fake stationary rough Heston model highlight the impact of rough volatilities and stochastic correlations on the optimal Markowitz strategies.