arXiv subjects

Serena Della Corte

Publications and source records attributed to Serena Della Corte.

9 recordsLinked to original sources

arXiv2026

A Strict Comparison Principle for Integro-Differential Hamilton-Jacobi-Bellman Equations on Domains with Boundary

This work provides a comparison principle for viscosity solutions to boundary value problems on (partially) bounded, cylindrical spaces. The comparison principle is based on a test function framework, that allows for the simultaneous treatment of diffusive as well as jump terms. Estimates in the proof of the comparison principle incorporate the use of Lyapunov functions that act as growth bounds for the solutions, effectively yielding a theory for unbounded viscosity solutions. We apply the results to a wide class of parabolic equations and elliptic problems on a space with corners.

math.AP↗

arXiv2026

Sample-path Large deviations for Scheduled Arrival Processes with Unpunctuality

We study a scheduled-arrival process in which deterministic arrival times are perturbed by i.i.d. unpunctuality random variables. For the empirical mean of independent copies of the resulting counting process, we prove a sample-path large deviations principle on $D([0,T])$, equipped with the Skorokhod topology. As an application, we derive large deviation principles for the corresponding workload process under deterministic and random service setting. In the deterministic service case, we also identify the exponentially tilted arrival path associated with a rare workload overflow event.

math.PR↗

arXiv2026

Machine learning for option pricing: an empirical investigation of network architectures

We consider the supervised learning problem of learning the price of an option or the implied volatility given appropriate input data (model parameters) and corresponding output data (option prices or implied volatilities). The majority of articles in this literature considers a (plain) feed forward neural network architecture in order to connect the neurons used for learning the function mapping inputs to outputs. In this article, motivated by methods in image classification and recent advances in machine learning methods for PDEs, we investigate empirically whether and how the choice of network architecture affects the accuracy and training time of a machine learning algorithm. We find that the generalized highway network architecture achieves the best performance, when considering the mean squared error and the training time as criteria, within the considered parameter budgets for the Black-Scholes and Heston option pricing problems. Considering the transformed implied volatility problem, a simplified DGM variant achieves the lowest error among the tested architectures. We also carry out a capacity-normalised comparison for completeness, where all architectures are evaluated with an equal number of parameters. Finally, for the implied volatility problem, we additionally include experiments using real market data.

q-fin.CP↗

arXiv2025

Existence of viscosity solutions for Hamilton-Jacobi equations via Lyapunov control

We give a new perspective on the existence of viscosity solutions for a stationary and a time-dependent first-order Hamilton-Jacobi equation. Following recent comparison principles, we work in a framework in which we consider a subsolution and a supersolution for two equations in terms of two Hamiltonians that can be seen as an upper semi-continuous upper and lower semi-continuous lower bound of our original Hamiltonian respectively. The bounds are made rigorous in terms of Youngs inequality. The bounds are furthermore formulated in a way that incorporate a Lyapunov function which allows us to restrict part of the analysis to compact sets and to work with almost optimizers of the considered control problems. For this reason, we can relax assumptions on the control problem: most notably, we do not need completeness of set of controlled paths. Moreover, this strategy avoids a-priori analysis on the regularity of the candidate solutions. To complete our picture, we exhibit our result in two contexts. First, we consider Riemannian manifolds with smooth boundary, in which the dynamics allows for both "inward" and "outward" drift. The boundary conditions are embedded into the Hamiltonians itself. Second, we consider the solutions to a Fillipov differential equation, i.e. one with discontinuous vector field. We show that our notion of Hamiltonians leverages the natural embedding of the discontinuity in an associated set-valued differential inclusion.

math.AP↗

arXiv2024

A comparison principle based on couplings of partial integro-differential operators

This paper is concerned with a comparison principle for viscosity solutions to Hamilton-Jacobi (HJ), -Bellman (HJB), and -Isaacs (HJI) equations for general classes of partial integro-differential operators. Our approach innovates in three ways: (1) We reinterpret the classical doubling-of-variables method in the context of second-order equations by casting the Ishii-Crandall Lemma into a test function framework. This adaptation allows us to effectively handle non-local integral operators, such as those associated with Lévy processes. (2) We translate the key estimate on the difference of Hamiltonians in terms of an adaptation of the probabilistic notion of couplings, providing a unified approach that applies to differential, difference, and integral operators. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology. We apply our theory to a variety of examples, in particular, to second-order differential operators and, more generally, generators of spatially inhomogeneous Lévy processes.

math.AP↗

arXiv2024

Uniform Sobolev, interpolation and geometric Calderón-Zygmund inequalities for graph hypersurfaces

In this note, our aim is to show that families of smooth hypersurfaces of $\mathbb R^{n+1}$ which are all $C^1$--close enough to a fixed compact, embedded one, have uniformly bounded constants in some relevant inequalities for mathematical analysis, like Sobolev, Gagliardo-Nirenberg and ``geometric'' Calderón-Zygmund inequalities. This technical result is quite useful, in particular, in the study of the geometric flows of hypersurfaces.

math.DG↗

arXiv2024

Large deviations for Markov processes with switching and homogenisation via Hamilton-Jacobi-Bellman equations

We consider the context of molecular motors modelled by a diffusion process driven by the gradient of a weakly periodic potential that depends on an internal degree of freedom. The switch of the internal state, that can freely be interpreted as a molecular switch, is modelled as a Markov jump process that depends on the location of the motor. Rescaling space and time, the limit of the trajectory of the diffusion process homogenizes over the periodic potential as well as over the internal degree of freedom. Around the homogenized limit, we prove the large deviation principle of trajectories with a method developed by Feng and Kurtz based on the analysis of an associated Hamilton--Jacobi--Bellman equation with an Hamiltonian that here, as an innovative fact, depends on both position and momenta.

math.PR↗

arXiv2023

Well-posedness of a Hamilton-Jacobi-Bellman equation in the strong coupling regime

We prove comparison principle for viscosity solutions of a Hamilton-Jacobi-Bellman equation in a strong coupling regime considering a stationary and a time-dependent version of the equation. We consider a Hamiltonian that has a representation as the supremum of a difference of two functions: an internal Hamiltonian depending on a control variable and a function interpreted as a cost of applying the controls. Our major innovation lies in the use of a cost function that can be discontinuous, unbounded and depending on momenta, enabling us to address previously unexplored scenarios such as cases arising from the theory of large deviations and homogenisation. For completeness, we also state the existence of viscosity solutions and we verify the assumptions for an example arising from biochemistry.

math.AP↗

arXiv2022

Global Existence and Stability for the Modified Mullins-Sekerka and Surface Diffusion Flow

In this survey we present the state of the art about the asymptotic behavior and stability of the modified Mullins--Sekerka flow and the surface diffusion flow of smooth sets, mainly due to E.~Acerbi, N.~Fusco, V.Julin and M.Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the strict stability property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under $W^{2,p}$-perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently "close" to a smooth strictly stable critical set $E$, both flows exist for all positive times and asymptotically "converge" to a translate of $E$.

math.AP↗