arXiv subjects

Gabriele Bolli

Publications and source records attributed to Gabriele Bolli.

4 recordsLinked to original sources

arXiv2026

Continuous Differentiability of the Value Function for Infinite-Dimensional Finite-Horizon Optimal Stopping and Related Variational Inequalities

This paper studies finite-horizon optimal stopping problems for semilinear stochastic evolution equations in real, separable Hilbert spaces, together with their associated parabolic variational inequalities. We prove continuous differentiability of the value function in infinite dimensions, thereby obtaining a smooth-fit principle for the corresponding optimal stopping problem. The analysis has two parts. First, we prove existence and uniqueness, in a suitable weighted class, of a mild solution to the variational inequality and identify it with the optimal stopping value function. We also establish local spatial Lipschitz continuity of the value function by probabilistic methods, without requiring any smoothing property of the underlying transition semigroup. Second, under a global regularizing assumption on this semigroup, we prove higher-order spatial regularity: the value function is continuously Fr\'echet differentiable and its gradient is locally H\"older continuous. The abstract results are then applied to a stochastic heat equation with additive noise.

math.PR↗

arXiv2026

Projected Evolutionary Lifting and Well-Posedness of Stationary Hamilton-Jacobi-Bellman Equations in Infinite Dimensions

This paper establishes the existence and uniqueness of mild solutions to stationary Hamilton-Jacobi-Bellman (HJB) equations associated with infinite-horizon stochastic optimal control problems in separable Hilbert spaces. Our framework includes settings with a lack of global smoothing properties of the transition semigroup, singular dynamics involving unbounded control operators, and state-dependent running costs. We overcome these challenges by lifting the state space using the Projected Evolutionary Lifting technique. This work is an extension of G. Bolli and F. Gozzi, Lifting and partial smoothing for stationary HJB equations and related control problems in infinite dimensions, 2025, in which existence and uniqueness is proved via a contraction mapping argument and is consequently restricted to sufficiently large discount factors. We remove this restriction, proving existence and uniqueness for any discount rate $\lambda > 0$ using tools from the theory of maximally monotone operators.

math.OC↗

arXiv2026

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 $\Gamma$, which we call $\Gamma$-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.

math.OC↗

arXiv2025

Lifting and partial smoothing for stationary HJB equations and related control problems in infinite dimensions

We study a family of stationary Hamilton-Jacobi-Bellman (HJB) equations in Hilbert spaces arising from stochastic optimal control problems. The main difficulties to treat such problems are: the lack of smoothing properties of the linear part of the HJB equation; the presence of unbounded control operators; the presence of state-dependent costs. This features, combined together, prevent the use of the classical mild solution theory of HJB equation (see e.g., Chapter 4 of G. Fabbri, F. Gozzi, A. Swiech, Stochastic Optimal Control in Infinite Dimensions: Dynamic Programming and HJB Equations, Springer, 2017). The problem has been studied in the evolutionary case in F. Gozzi, F. Masiero, Lifting Partial Smoothing to Solve HJB Equations and Stochastic Control Problems, SIAM Journal on Control and Optimization, 63(3), (2025), pp. 1515-1559 using a "lifting technique" (i.e. working in a suitable space of trajectories where a "partial smoothing" property of the linear part of the HJB equations holds. In this paper we extend such a theory to the case of infinite horizon optimal control problems, which are very common, in particular in economic applications. The main results are: the existence and uniqueness of a regular mild solution to the HJB equation; a verification theorem, and the synthesis of optimal feedback controls.

math.OC↗