Optimal control of stochastic Volterra integral equations with completely monotone kernels and stochastic differential equations on Hilbert spaces with unbounded control and diffusion operators

The dynamic programming approach is one of the most powerful ones in optimal control. However, when dealing with optimal control problems of stochastic Volterra integral equations (SVIEs) with completely monotone kernels, deep mathematical difficulties arise and it is still not understood. These very classical problems have applications in most fields and have now become even more popular due to their applications in mathematical finance under rough volatility. In this article, we consider a class of optimal control problems of SVIEs with completely monotone kernels. Via a recent Markovian lift \cite{FGW2024}, the problem can be reformulated as an optimal control problem of stochastic differential equations (SDEs) on suitable Hilbert spaces, which due to the roughness of the kernel, presents a generator of an analytic semigroup and unbounded control and diffusion operators. This analysis leads us to study a general class of optimal control problems of abstract SDEs on Hilbert spaces with unbounded control and diffusion operators. This class includes optimal control problems of SVIEs with completely monotone kernels, but it is also motivated by other models. We analyze the regularity of the associated Ornstein-Uhlenbeck transition semigroup. We prove that the semigroup exhibits a new smoothing property in control directions through a general observation operator $Γ$, which we call $Γ$-smoothing. This allows us to establish existence and uniqueness of mild solutions of the Hamilton-Jacobi-Bellman equation, establish a verification theorem, and construct optimal feedback controls. We apply these results to optimal control problems of SVIEs with completely monotone kernels. To the best of our knowledge these are the first results of this kind for this abstract class of infinite dimensional problems and for the optimal control of SVIEs with completely monotone kernels.

💡 Research Summary

The paper tackles the optimal control of stochastic Volterra integral equations (SVIEs) whose kernels are completely monotone—a class of models that has recently attracted great interest, especially in rough‑volatility finance. The main difficulty lies in the fact that such SVIEs are non‑Markovian and involve memory effects that prevent the direct use of dynamic programming. The authors overcome this obstacle by employing a recent Markovian lift (FGW 2024) which embeds the original SVIE into an abstract stochastic differential equation (SDE) on a suitably chosen Hilbert space (H). In this lifted formulation the state evolves according to

\