Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Gabriele Bolli, Filippo de Feo

Abstract

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.

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

Abstract

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 , which we call -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.
Paper Structure (42 sections, 16 theorems, 98 equations)

This paper contains 42 sections, 16 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Model problems
  3. Optimal control of SVIEs with completely monotone kernels.
  4. Optimal control of SDEs on Hilbert spaces with unbounded control and diffusion operators.
  5. State of the art
  6. Optimal control of SDEs in infinite dimensions.
  7. Optimal control of SVIEs.
  8. Our results
  9. Optimal control of SDEs on Hilbert spaces with unbounded control and diffusion operators.
  10. Optimal control of SVIEs with completely monotone kernels.
  11. The abstract stochastic optimal control problem
  12. Notations.
  13. Abstract framework.
  14. State equation.
  15. Cost functional.
  16. ...and 27 more sections

Key Result

Theorem 3.4

Let $\phi\in B_{b}^{\Gamma}(H)$ with $\Gamma \in \mathcal{L}(H, \mathcal{Y})$ and assume Hypothesis hyp:partial_smoothing holds. Then, for any $t>0$, $P_{t}[\phi]$ is differentiable in the generalized directions of $\operatorname{Im}(B)$. Specifically, for any $x \in \overline{H}$, there exists a ve Here, the term $\Gamma y$ is understood in the sense of the projected Gaussian measure on $\mathcal

Theorems & Definitions (52)

  • Definition 2.2: Mild Solution
  • Definition 3.1: $\Gamma$-Cylindrical Functions
  • Remark 3.2: Well-posedness of the evaluation on $\overline{H}$
  • Theorem 3.4: $B$-Differentiability via $\Gamma$-regularization
  • proof
  • Theorem 4.1: Minimum Energy Characterization
  • proof
  • Definition 5.1: Smoothing Spaces on $\overline{H}$
  • Remark 5.2
  • Lemma 5.3: Preservation of Regularity on $\overline{H}$
  • ...and 42 more