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Mild solutions of HJB equations associated with cylindrical stable Lévy noise in infinite dimensions

Alessandro Bondi, Fausto Gozzi, Enrico Priola, Jerzy Zabczyk

TL;DR

We address optimal control of an infinite-dimensional system driven by cylindrical α-stable noise and derive a nonlocal, parabolic Hamilton–Jacobi–Bellman equation. Using the dynamic programming framework and the smoothing properties of the Ornstein–Uhlenbeck semigroup, we establish the existence and uniqueness of a mild solution in the function space $C^1_\gamma(H)$ and prove quantitative regularity results for the spatial gradient $Du$, including Hölder continuity. The explicit Hamiltonian $H(p)$ is derived from a quadratic control cost, enabling a mild formulation $u(t,x)=P_t h(x)+\int_0^t P_{t-s}[\mathcal{H}(\cdot,Du(s,\cdot))](x) ds$ without relying on viscosity solutions. These results lay the groundwork for a Verification Theorem and extend stochastic control methods to infinite-dimensional systems with pure jump noise.

Abstract

We study the optimal control of an infinite-dimensional stochastic system governed by an SDE in a separable Hilbert space driven by cylindrical stable noise. We establish the existence and uniqueness of a mild solution to the associated HJB equation. This result forms the basis for the proof of the Verification Theorem, which is the subject of ongoing research and will provide a sufficient condition for optimality.

Mild solutions of HJB equations associated with cylindrical stable Lévy noise in infinite dimensions

TL;DR

We address optimal control of an infinite-dimensional system driven by cylindrical α-stable noise and derive a nonlocal, parabolic Hamilton–Jacobi–Bellman equation. Using the dynamic programming framework and the smoothing properties of the Ornstein–Uhlenbeck semigroup, we establish the existence and uniqueness of a mild solution in the function space and prove quantitative regularity results for the spatial gradient , including Hölder continuity. The explicit Hamiltonian is derived from a quadratic control cost, enabling a mild formulation without relying on viscosity solutions. These results lay the groundwork for a Verification Theorem and extend stochastic control methods to infinite-dimensional systems with pure jump noise.

Abstract

We study the optimal control of an infinite-dimensional stochastic system governed by an SDE in a separable Hilbert space driven by cylindrical stable noise. We establish the existence and uniqueness of a mild solution to the associated HJB equation. This result forms the basis for the proof of the Verification Theorem, which is the subject of ongoing research and will provide a sufficient condition for optimality.

Paper Structure

This paper contains 6 sections, 5 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminary material
  3. On stable Lévy processes in Hilbert spaces
  4. The State Equation
  5. The Control Problem and the associated HJB equation
  6. Mild solutions of the HJB equation

Key Result

Lemma 2.2

For every $a\in \mathcal{U}$, $t\ge0$ and $x\in H$, ab1 admits a unique predictable mild solution with $p-$locally integrable paths for $p\in[1,\alpha)$, that is, there exists a unique predictable process $X=(X_s)_{s\ge t}$ with trajectories in $L^p_{\text{loc}}(s,\infty)$ for any $p\in[1,\alpha)$ s

Theorems & Definitions (11)

  • Remark 1.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 3.1
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • ...and 1 more