Utility Maximization in Multivariate Volterra Models

Abstract

This paper is concerned with portfolio selection for an investor with power utility in multiasset financial markets in a rough stochastic environment. We investigate Merton’s portfolio problem for different multivariate Volterra models, covering the rough Heston model. First we consider a class of multivariate affine Volterra models introduced in [E. Abi Jaber, E. Miller, and H. Pham, SIAM J. Financial Math.., 12 (2021), pp. 369–409]. Based on the classical Wishart model described in [N. Bäuerle and Z. Li, J. Appl. Probab., 50 (2013), pp. 1025–1043], we then introduce a new matrix-valued stochastic volatility model, where the volatility is driven by a Volterra–Wishart process. Due to the non-Markovianity of the underlying processes, the classical stochastic control approach cannot be applied in these settings. To overcome this issue, we provide a verification argument using calculus of convolutions and resolvents. The resulting optimal strategy can then be expressed explicitly in terms of the solution of a multivariate Riccati–Volterra equation. We thus extend 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 [B. Han and H. Y. Wong, Finance Res. Lett., 39 (2021)]. We also provide existence and uniqueness theorems for the occurring Volterra processes and illustrate our results with a numerical study.