On explicit solutions to a class of quadratic BSDEJs driven by affine Volterra processes with jumps and applications

Abstract

In this paper we consider a class of quadratic BSDEs with jumps (quadratic BSDEJs) involving inhomogeneous affine Volterra processes and show that their solution can be reduced to solving a system of generalized inhomogeneous integral Riccati-Volterra ordinary differential equations with L\'evy jump compensators. This yields a rich and flexible class of quadratic BSDEJs that are analytically tractable, in the sense that their solutions are explicit up to the solution of an associated integral Riccati-Volterra ODE with L\'evy jump compensator. As an application, we provide analytically tractable solutions to the continuous-time Markowitz mean-variance portfolio selection problem within a multivariate class of affine Volterra models allowing jumps driven by an independent Poisson random measure. 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 the martingale optimality principle by constructing a family of submartingale processes characterized via solutions to a novel Riccati backward stochastic differential equation with jumps (Riccati BSDEJ), particular subclass of the aforementionned quadratic BSDEJ. Specifically, we obtain analytical closed-form expressions for the optimal feedback control as well as the mean-variance efficient frontier, both of which depend on the solution to the associated multivariate inhomogeneous Riccati-Volterra system, while the optimal value function is expressed using the solution to this original Riccati BSDEJ. Furthermore, numerical experiments on a two-dimensional fake stationary rough Heston model is discussed and used to highlight the impact of stabilized rough volatilities on the Markowitz allocation problem.