Optimal Merton's Problem under Multivariate Affine Volterra Models with Jumps

TL;DR

This paper addresses optimal portfolio selection in complex multi-asset markets with jumps and rough volatility, introducing novel Riccati BSDEJ solutions for non-Markovian, non-semimartingale models.

Contribution

It develops a new approach using Riccati BSDEJs to solve Merton's problem in multivariate Volterra models with jumps, extending classical methods to non-Markovian settings.

Findings

Optimal strategies are derived in semi-closed form depending on Riccati-Volterra equations.

Numerical experiments show the effects of path roughness and jumps on the value function.

The approach handles non-Markovian, non-semimartingale models with jumps effectively.

Abstract

This paper is concerned with portfolio selection for an investor with exponential, power, and logarithmic utility in multi-asset financial markets allowing jumps. We investigate the classical Merton's portfolio optimization problem in a Volterra stochastic environment described by a multivariate Volterra--Heston model with jumps driven by an independent Poisson random measure. Owing to the non-Markovian and non-semimartingale nature of the model, classical stochastic control techniques are not directly applicable. Instead, the problem is tackled using the martingale optimality principle by constructing a family of supermartingale processes characterized via solutions to an original Riccati backward stochastic differential equation with jumps (Riccati BSDEJ).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 BSDEJ. Numerical experiments on a two-dimensional rough Heston model illustrate the impact of both path roughness and jumps components on the value function and optimal strategies in the Merton problem.