arXiv subjects

Stefano Pagliarani

Publications and source records attributed to Stefano Pagliarani.

At least 19 recordsLinked to original sources

arXiv2026

McKean-Vlasov Differential Equations: An introduction and a focus on some kinetic models

These notes were prepared for a series of lectures delivered at the 43rd Finnish Summer School on Probability and Statistics, held in Lammi, Finland, from May 26 to 30, 2025. They provide an introduction to McKean-Vlasov stochastic differential equations and their connections with non-linear Fokker-Planck equations and mean-field interacting particle systems. Particular attention is devoted to density-dependent equations and to kinetic-type models with degenerate noise. For the latter, we introduce the underlying non-Euclidean geometry and the associated anisotropic and intrinsic Hölder spaces, and present well-posedness and regularity results based on Schauder estimates and superposition principles.

math.PR↗

arXiv2026

On Heat kernel Estimtes for Brownian SDEs with Distributional Drift

We establish heat-kernel bounds and regularity estimates for the transition densities of the diffusion associated with the martingale problem corresponding to the generator of a formal multidimensional Brownian SDE with singular drift. As a by-product, we also derive Schauder estimates for the associated Kolmogorov (kinetic) Cauchy problem. We consider both the cases of non-degenerate and degenerate noise (e.g. kinetic-type models), in the so-called Young regime. Namely, we consider a time inhomogeneous drift in L $\infty$ [0,T ] C $β$ o for some fixed time horizon T , where ), with o standing for an underlying distance, namely the usual Euclidean one in the non degenerate setting, and the scale-homogeneous one in the kinetic case. Importantly, the estimates are obtained by employing as parametrix the transition density of the SDE (with variable coefficients) without singular perturbation, as opposed to the standard Levi parametrix obtained by freezing the noise. Finally, since the noise is multiplicative, the weak well-posedness of the singular SDE is a novel result in itself, and the density estimates directly imply irreducibility and strong Feller property of its solutions.

math.AP↗

arXiv2026

Nash equilibrium in a singular stochastic game between two renewable power producers with price impact

We study the singular stochastic game, formulated in Awerkin and Vargiolu (Decis. Econ. Finance 44(2), 2021), between two agents aiming at maximizing their profits by installing photovoltaic panels and selling the produced electricity, net of installation costs, in the case that their cumulative installations have an impact on power prices. We first solve explicitly the static, one-step, version of the game, and find that Nash equilibria divide the state-space into four regions: one where both players are idle, two where only one player installs new panels, and one where both players install. In some particular regimes, we find that the latter may not be uniquely distinguished from the previous two. We then consider the dynamic, continuous-time, problem. Led by the intuition garnered in the static case, we assume a free-boundary structure similar to that arising in the one-step game and provide a rigorous verification theorem for the corresponding system of free-boundary HJB equations, also taking into account the lack of smoothness of the value functions near the free boundaries. Finally, for each regular solution of the HJB system, we show that there exists a unique equilibrium strategy, which is obtained as the solution to the Skorokhod-type problem associated with the free boundary.

math.OC↗

arXiv2026

Degenerate McKean-Vlasov equations with drift in anisotropic negative Besov spaces

The paper is concerned with a McKean-Vlasov type SDE with drift in anisotropic Besov spaces with negative regularity and with degenerate diffusion matrix under the weak H{ö}rmander condition. The main result is of existence and uniqueness of a solution in law for the McKean-Vlasov equation, which is formulated as a suitable martingale problem. All analytical tools needed are derived in the paper, such as the well-posedness of the Fokker-Planck and Kolmogorov PDEs with distributional drift, as well as continuity dependence on the coefficients. The solutions to these PDEs naturally live in anisotropic Besov spaces, for which we developed suitable analytical inequalities, such as Schauder estimates.

math.PR↗

arXiv2026

Numerical approximation of McKean-Vlasov SDEs via stochastic gradient descent

We propose a novel approach to numerically approximate McKean-Vlasov stochastic differential equations (MV-SDE) using stochastic gradient descent (SGD) while avoiding the use of interacting particle systems (IPS) {and the associated simulation costs required to achieve the ``propagation of chaos'' limit}. The SGD technique is deployed to solve a Euclidean minimization problem, obtained by first representing the MV-SDE as a minimization problem over the set of continuous functions of time, and then approximating the domain with a finite-dimensional subspace. Convergence is established by proving certain intermediate stability and moment estimates of the relevant stochastic processes, including the tangent processes. Numerical experiments illustrate the competitive performance of our SGD based method compared to the IPS benchmarks. This work offers a theoretical foundation for using the SGD method in the context of numerical approximation of MV-SDEs, and provides analytical tools to study its stability and convergence.

math.NA↗

arXiv2025

Strong regularization by noise for a class of kinetic SDEs driven by symmetric α-stable processes

We establish strong well-posedness for a class of degenerate SDEs of kinetic type with autonomous diffusion driven by a symmetric $α$-stable process under Hölder regularity conditions for the drift term. We partially recover the thresholds for the Hölder regularity that are optimal for weak uniqueness. In general dimension, we only consider $α= 2$ and need an additional integrability assumption for the gradient of the drift: this condition is satisfied by Peano-type functions. In the one-dimensional case we do not need any additional assumption. In the multi-dimensional case, the proof is based on a first-order Zvonkin transform/PDE, while for the one-dimensional case we use a second-order Zvonkin/PDE transform together with a Watanabe-Yamada technique.

math.PR↗

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

Intrinsic Hölder spaces for fractional kinetic operators

We introduce anisotropic Hölder spaces useful for the study of the regularity theory for non local kinetic operators $\mathcal{L}$ whose prototypal example is \begin{equation} \mathcal{L} u (t,x,v) = \int_{\mathbb{R}^d} \frac{C_{d,s}}{|v - v'|^{d+2s}} (u(t,x,v') - u(t,x,v)) d v' + \langle v , \nabla_x \rangle + \partial_t, \quad (t,x,v)\in\mathbb{R}\times\mathbb{R}^{2d}. \end{equation} The Hölder spaces are defined in terms of an anisotropic distance relevant to the Galilean geometric structure on $\mathbb{R}\times\mathbb{R}^{2d}$ the operator $\mathcal{L}$ is invariant with respect to. We prove an intrinsic Taylor-like formula, whose reminder is estimated in terms of the anisotropic distance of the Galilean structure. Our achievements naturally extend analogous known results for purely differential operators on Lie groups.

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↗

arXiv2021

A Yosida's parametrix approach to Varadhan's estimates for a degenerate diffusion under the weak Hörmander condition

We adapt and extend Yosida's parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under the weak Hörmander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of path-dependent financial derivatives known as Asian options. We obtain the Varadhan formula \begin{equation} \frac{-2 \log p(t,x;T,y) } { Ψ(t,x;T,y) } \to 1, \qquad \text{as } \quad T-t \to 0^+, \end{equation} where $p$ denotes the transition density and $Ψ$ denotes the optimal cost function of a deterministic control problem associated to the diffusion. We provide a partial proof of this formula, and present numerical evidence to support the validity of an intermediate inequality that is required to complete the proof. We also derive an asymptotic expansion of the cost function $Ψ$, expressed in terms of elementary functions, which is useful in order to design efficient approximation formulas for the transition density.

math.PR↗

arXiv2018

A Fourier-based Picard-iteration approach for a class of McKean-Vlasov SDEs with Lévy jumps

We consider a class of Lévy-driven stochastic differential equations (SDEs) with McKean-Vlasov (MK-V) interaction in the drift coefficient. It is assumed that the coefficient is bounded, affine in the state variable, and only measurable in the law of the solution. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK-V SDE, and derive a discretized Picard iteration scheme that approximates the law of the solution through its characteristic function. Numerical illustrations show the effectiveness of our method, which appears to be appropriate to handle the multi-dimensional setting.

math.PR↗

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↗

arXiv2017

Intrinsic Taylor formula for non-homogeneous Kolmogorov-type Lie groups

We prove an intrinsic Taylor-like formula for a class of Lie groups arising in the study of some sub-elliptic differential operators, namely the Kolmogorov operators. The estimate of the remainder is in terms of the intrinsic norm induced by such operators. These results extend the recent developments in a work by Pascucci and the present authors, where a full characterization of the intrinsic Hölder spaces and their Taylor polynomials were given under the additional assumption that the Lie group is homogeneous in the sense of Folland & Stein. Remarkably, the intrinsic Taylor polynomial admits the same representation as in the homogeneous case.

math.AP↗

arXiv2016

The exact Taylor formula of the implied volatility

In a model driven by a multi-dimensional local diffusion, we study the behavior of implied volatility σ and its derivatives with respect to log-strike k and maturity T near expiry and at the money. We recover explicit limits of these derivatives for (T,k) approaching the origin within the parabolic region |x-k|^2 < λ T, with x denoting the spot log-price of the underlying asset and where λ is a positive and arbitrarily large constant. Such limits yield the exact Taylor formula for implied volatility within the parabola |x-k|^2 < λ T. In order to include important models of interest in mathematical finance, e.g. Heston, CEV, SABR, the analysis is carried out under the assumption that the infinitesimal generator of the diffusion is only locally elliptic.

math.PR↗

arXiv2015

Intrinsic Taylor formula for Kolmogorov-type homogeneous groups

We consider a class of ultra-parabolic Kolmogorov-type operators satisfying the Hormander's condition. We prove an intrinsic Taylor formula with global and local bounds for the remainder given in terms of the norm in the homogeneous Lie group naturally associated to the differential operator.

math.AP↗