Searcharxiv⌕ Search

arXiv subjects

Andrea Pascucci

Publications and source records attributed to Andrea Pascucci.

At least 19 recordsLinked to original sources
arXiv2026

Trading in residential energy systems with storage: a kinetic mean-field approach

We study a stochastic optimal control problem motivated by the operation of a large ensemble of residential storage devices coordinated by an energy aggregator. The aggregator remunerates prosumers in exchange for direct control of their batteries and seeks to jointly (i) reduce local supply-demand imbalances and (ii) exploit intraday price fluctuations through energy arbitrage. The core modeling feature is a kinetic mean-field formulation: the state of charge is treated as a position, the charging/discharging power as a velocity, and the control as an acceleration, thus encoding ramp-rate limitations and producing smooth power trajectories. This leads to a controlled McKean-Vlasov Langevin-type system in which both the drift and the objective functional depend on the time-marginal law of the state, allowing one to capture endogenous interaction effects and population-level stabilization incentives. The performance criterion combines the cost of grid exchange with convex penalties representing degradation and control effort, and includes mean-field terms that promote alignment with the population average; terminal contributions account for residual energy value and end-of-horizon coordination. The resulting control problem is Markovian and hypoelliptic, and naturally connects mean-field control with ultraparabolic operators of kinetic type. This viewpoint provides a coherent bridge between physically constrained storage actuation and law-dependent incentives in large-scale energy management. Numerical experiments based on deep learning solvers are presented to validate the model. From a computational standpoint, the problem is particularly challenging, as it yields a fully coupled forward-backward stochastic system associated with a five-dimensional Hamilton-Jacobi-Bellman equation.

math.OC↗
arXiv2025

Well-posedness of kinetic McKean-Vlasov equations

We consider the McKean-Vlasov equation $dX_t = b(t, X_t, [X_t])dt + σ(t, X_t, [X_t])dW_t$ where $[X_t]$ is the law of $X_t$. We specifically consider the kinetic case, where the equation is degenerate because the dimension of the Brownian motion $W$ is strictly smaller than that of the solution $X$, as commonly required in classical models of collisional kinetic theory. Assuming Hölder continuous coefficients and a weak Hörmander condition, we prove the well-posedness of the equation. This result advances the existing literature by filling a crucial gap: it addresses the previously unexplored case where the diffusion coefficient $σ$ depends on the law $[X_t]$. Notably, our proof employs a simplified and direct argument eliminating the need for PDEs involving derivatives with respect to the measure argument. A critical ingredient is the sub-Riemannian metric structure induced by the corresponding Fokker-Planck operator.

math.PR↗
arXiv2024

McKean-Vlasov stochastic equations with Hölder coefficients

This work revisits the well-posedness of non-degenerate McKean-Vlasov stochastic differential equations with Hölder continuous coefficients, recently established by Chaudru de Raynal. We provide a streamlined and direct proof that leverages standard Gaussian estimates for uniformly parabolic PDEs, bypassing the need for derivatives with respect to the measure argument and extending applicability to hypoelliptic PDEs under weaker assumptions.

math.PR↗
arXiv2024

Sobolev embeddings for kinetic Fokker-Planck equations

We introduce intrinsic Sobolev-Slobodeckij spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak Hörmander condition. We prove continuous embeddings into Lorentz and intrinsic Hölder spaces. We also prove approximation and interpolation inequalities by means of an intrinsic Taylor expansion, extending analogous results for Hölder spaces. The embedding at first order is proved by adapting a method by Luc Tartar which only exploits scaling properties of the intrinsic quasi-norm, while for higher orders we use uniform kernel estimates.

math.AP↗
arXiv2024

Optimal Schauder estimates for kinetic Kolmogorov equations with time measurable coefficients

We prove global Schauder estimates for kinetic Kolmogorov equations with coefficients that are Hölder continuous in the spatial variables but only measurable in time. Compared to other available results in the literature, our estimates are optimal in the sense that the inherent Hölder spaces are the strongest possible under the given assumptions: in particular, under a parabolic Hörmander condition, we introduce Hölder norms defined in terms of the intrinsic geometry that the operator induces on the space-time variables. The technique is based on the existence and the regularity estimates of the fundamental solution of the equation. These results are essential for studying backward Kolmogorov equations associated with kinetic-type diffusions, e.g. stochastic Langevin equation.

math.AP↗
arXiv2023

Optimal regularity for degenerate Kolmogorov equations with rough coefficients

We consider a class of degenerate equations satisfying a parabolic Hörmander condition, with coefficients that are measurable in time and Hölder continuous in the space variables. By utilizing a generalized notion of strong solution, we establish the existence of a fundamental solution and its optimal Hölder regularity, as well as Gaussian estimates. These results are key to study the backward Kolmogorov equations associated to a class of Langevin-type diffusions.

math.AP↗
arXiv2022

Strong regularization by noise for kinetic SDEs

In this paper we prove strong well-posedness for a system of stochastic differential equations driven by a degenerate diffusion satisfying a weak-type Hörmander condition, assuming Hölder regularity assumptions on the drift coefficient. This framework encompasses, as particular cases, stochastic Langevin systems of kinetic SDEs. The drift coefficient of the velocity component is allowed to be $α$-Hölder continuous without any restriction on the index $α$, which can be any positive number in $]0,1[$. As the deterministic counterparts of these differential systems are not well-posed, this result can be viewed as a phenomenon known as regularization by noise.

math.PR↗
arXiv2022

Numerical solution of kinetic SPDEs via stochastic Magnus expansion

In this paper, we show how the Itô-stochastic Magnus expansion can be used to efficiently solve stochastic partial differential equations (SPDE) with two space variables numerically. To this end, we will first discretize the SPDE in space only by utilizing finite difference methods and vectorize the resulting equation exploiting its sparsity. As a benchmark, we will apply it to the case of the stochastic Langevin equation with constant coefficients, where an explicit solution is available, and compare the Magnus scheme with the Euler-Maruyama scheme. We will see that the Magnus expansion is superior in terms of both accuracy and especially computational time by using a single GPU and verify it in a variable coefficient case. Notably, we will see speed-ups of order ranging form 20 to 200 compared to the Euler-Maruyama scheme, depending on the accuracy target and the spatial resolution.

math.NA↗
arXiv2022

On the stochastic Magnus expansion and its application to SPDEs

We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time τ and provide a novel asymptotic estimate of the cumulative distribution function of t. As an application, we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature of the method is that it is fully parallelizable. We also present numerical tests in order to asses the accuracy of the numerical schemes.

math.PR↗
arXiv2022

A probabilistic result on impulsive noise reduction in Topological Data Analysis through Group Equivariant Non-Expansive Operators

In recent years, group equivariant non-expansive operators (GENEOs) have attracted attention in the fields of Topological Data Analysis and Machine Learning. In this paper we show how these operators can be of use also for the removal of impulsive noise and to increase the stability of TDA in the presence of noisy data. In particular, we prove that GENEOs can control the expected value of the perturbation of persistence diagrams caused by uniformly distributed impulsive noise, when data are represented by $L$-Lipschitz functions from $\mathbb{R}$ to $\mathbb{R}$.

cs.CG↗
arXiv2021

Backward and forward filtering under the weak Hörmander condition

We derive the forward and backward filtering equations for a class of degenerate partially observable diffusions, satisfying the weak Hörmander condition. Our approach is based on the Hölder theory for degenerate SPDEs that allows to pursue the direct approaches proposed by N. V. Krylov and A. Zatezalo, and A. Yu. Veretennikov, avoiding the use of general results from filtering theory. As a by-product we also provide existence, regularity and estimates for the filtering density.

math.PR↗
arXiv2019

The parametrix method for parabolic SPDEs

We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the Ito-Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.

math.PR↗
arXiv2019

On stochastic Langevin and Fokker-Planck equations: the two-dimensional case

We prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution of the stochastic Langevin equation. This degenerate SPDE satisfies the weak Hörmander condition. We use a Wentzell's transform to reduce the SPDE to a PDE with random coefficients; then we apply a new method, based on the parametrix technique, to construct a fundamental solution. This approach avoids the use of the Duhamel's principle for the SPDE and the related measurability issues that appear in the stochastic framework. Our results are new even for the deterministic equation.

math.PR↗
arXiv2019

Efficient Computation of Various Valuation Adjustments Under Local Lévy Models

Various valuation adjustments, or XVAs, can be written in terms of non-linear PIDEs equivalent to FBSDEs. In this paper we develop a Fourier-based method for solving FBSDEs in order to efficiently and accurately price Bermudan derivatives, including options and swaptions, with XVA under the flexible dynamics of a local Lévy model: this framework includes a local volatility function and a local jump measure. Due to the unavailability of the characteristic function for such processes, we use an asymptotic approximation based on the adjoint formulation of the problem.

q-fin.MF↗
arXiv2018

Local densities for a class of degenerate diffusions

We study a class of R^d-valued continuous strong Markov processes that are generated, only locally, by an ultra-parabolic operator with coefficients that are regular w.r.t. the intrinsic geometry induced by the operator itself and not w.r.t. the Euclidean one. The first main result is a local Ito formula for functions that are not twice-differentiable in the classical sense, but only intrinsically w.r.t. to a set of vector fields, related to the generator, satisfying the Hormander condition. The second main contribution, which builds upon the first one, is an existence and regularity result for the local transition density.

math.PR↗
arXiv2018

Systemic risk in a mean-field model of interbank lending with self-exciting shocks

In this paper we consider a mean-field model of interacting diffusions for the monetary reserves in which the reserves are subjected to a self- and cross-exciting shock. This is motivated by the financial acceleration and fire sales observed in the market. We derive a mean-field limit using a weak convergence analysis and find an explicit measure-valued process associated with a large interbanking system. We define systemic risk indicators and derive, using the limiting process, several law of large numbers results and verify these numerically. We conclude that self-exciting shocks increase the systemic risk in the network and their presence in interbank networks should not be ignored.

q-fin.MF↗