Utility maximization in multivariate Volterra models

Abstract

This paper is concerned with portfolio selection for an investor with power utility in multi-asset financial markets in a rough stochastic environment. We investigate Merton's portfolio problem for a class of multivariate affine Volterra models introduced in [5] and a Volterra-Wishart model based on the model described in [3], both covering the rough Heston model. Due to the non-Markovianity of the underlying processes the classical stochastic control approach can not be applied in this setting. To overcome this difficulty, we provide a verification argument inspired by [3] to show that an appropriate candidate is indeed the optimal portfolio strategy using calculus of convolutions and resolvents. The optimal strategy can then be expressed explicitly in terms of the solution of a multivariate Riccati-Volterra equation. This extends the results obtained by Han and Wong to the multivariate case, avoiding restrictions on the correlation structure linked to the martingale distortion transformation used in [9]. We also provide a numerical study to illustrate our results.