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Sara Mazzonetto

Publications and source records attributed to Sara Mazzonetto.

17 recordsLinked to original sources
arXiv2026

Sticky-threshold diffusions, local time approximation and parameter estimation

We study a class of high-frequency path functionals for one-dimensional diffusions with singular thresholds or boundaries, allowing for skewness, discontinuities in the diffusion coefficient, and stickiness or sticky reflection. These functionals, originally developed for local-time approximation in non-singular diffusions, are constructed from a test function and a diverging normalizing sequence. We establish convergence to local time, regardless of the nature of the singular threshold features, thereby extending several recent results on specific cases. Notably, our framework allows for any normalizing diverging sequence that is $o(n)$, where $n$ is the observation frequency, and thresholds at which several singular behaviors occur simultaneously. Combining the local-time approximation with occupation-time approximations, we construct consistent estimators of the stickiness and skewness parameters that remain valid in the presence of jump-discontinuities in the diffusion coefficient; this solves the estimation problem under the simultaneous presence of all these singular features.

math.PR↗
arXiv2026

Existence and uniqueness for singular stochastic differential equations with piecewise well-behaved coefficients

We study existence and uniqueness for one-dimensional generalized stochastic differential equations with singular coefficients, including distributional drift and degenerate, possibly discontinuous, diffusion coefficients. Such singularities naturally encode changes in the dynamics at thresholds, including reflecting, skew, or sticky interface behavior. We develop two directions. We provide sufficient conditions for pathwise uniqueness, under weak existence and uniqueness in law, without assuming uniform ellipticity or continuity of the diffusion coefficient. We also investigate a pasting approach for generalized stochastic differential equations that transfers strong existence and pathwise uniqueness, as well as weak existence and uniqueness in law, from local component equations to a global solution. To the best of our knowledge, this provides the first explicit pasting theorem yielding pathwise uniqueness in the setting of generalized stochastic differential equations. As an application, we establish the first existence and uniqueness results for a class of skew sticky threshold Cox-Ingersoll-Ross-type diffusions, including the threshold Chan-Karolyi-Longstaff-Sanders process.

math.PR↗
arXiv2025

On the number of crossings and bouncings of a diffusion at a sticky threshold

In this paper, we study the asymptotic behavior of the number of crossings by a one-dimensional diffusion of a threshold where the process exhibits stickiness. We distinguish three types of crossings and show that to each type corresponds a distinct asymptotic regime for the respective number of crossings statistic. We introduce notions of bouncing as the symmetric counterparts to crossings and show that the corresponding number of bouncings statistics share the same asymptotic properties as their crossings counterparts. We first prove the results for sticky Brownian motion, then extend them to sticky-reflected Brownian motion (where only bouncing is possible) and to sticky diffusions. As an application, we propose consistent estimators for the stickiness parameter of sticky diffusions and sticky-reflected Brownian motion.

math.PR↗
arXiv2025

Parameters estimation of a Threshold Chan-Karolyi-Longstaff-Sanders process from continuous and discrete observations

We consider a continuous time process that is self-exciting and ergodic, called threshold Chan-Karolyi-Longstaff-Sanders (CKLS) process. This process is a generalization of various models in econometrics, such as Vasicek model, Cox-Ingersoll-Ross, and Black-Scholes, allowing for the presence of several thresholds which determine changes in the dynamics. We study the asymptotic behavior of maximum-likelihood and quasi-maximum-likelihood estimators of the drift parameters in the case of continuous time and discrete time observations. We show that for high frequency observations and infinite horizon the estimators satisfy the same asymptotic normality property as in the case of continuous time observations. We also discuss diffusion coefficient estimation. Finally, we apply our estimators to simulated and real data to motivate considering (multiple) thresholds.

math.ST↗
arXiv2024

On the Itô-Alekseev-Gröbner formula for stochastic differential equations

In this article we establish a new formula for the difference of a test function of the solution of a stochastic differential equation and of the test function of an Itô process. The introduced formula essentially generalizes both the classical Alekseev-Gröbner formula from the literature on deterministic differential equations as well as the classical Itô formula from stochastic analysis. The proposed Itô-Alekseev-Gröbner formula is a powerful tool for deriving strong approximation rates for perturbations and approximations of stochastic ordinary and partial differential equations.

math.PR↗
arXiv2024

Beyond the delta method

We give an asymptotic development of the maximum likelihood estimator (MLE), or any other estimator defined implicitly, in a way which involves the limiting behavior of the score and its higher-order derivatives. This development, which is explicitly computable, gives some insights about the non-asymptotic behavior of the renormalized MLE and its departure from its limit. We highlight that the results hold whenever the score and its derivative converge, including to non Gaussian limits. Our approach is based on an asymptotic implicit function theorem, inspired from perturbative approaches.

math.ST↗
arXiv2024

Rates of convergence to the local time of Oscillating and Skew Brownian Motions

In this paper, a class of statistics based on high frequency observations of oscillating and skew Brownian motion is considered. Their convergence rate towards the local time of the underlying process is obtained in form of a functional limit theorem. Oscillating and skew Brownian motion are solutions to stochastic differential equations with singular coefficients: piecewise constant diffusion coefficient or additive local time finite variation term. The result is applied to provide estimators of the skewness parameter and study their asymptotic behavior, and diffusion coefficient estimation is discussed as well. Moreover, in the case of the classical statistics given by the normalized number of crossings, the result is proved to hold for a larger class of Itô processes with singular coefficients. Up to our knowledge, this is the first result proving the convergence rates for estimators of the skewness parameter of skew Brownian motion.

math.PR↗
arXiv2024

Estimation of parameters and local times in a discretely observed threshold diffusion model

We consider a simple mean reverting diffusion process, with piecewise constant drift and diffusion coefficients, discontinuous at a fixed threshold. We discuss estimation of drift and diffusion parameters from discrete observations of the process, with a generalized moment estimator and a maximum likelihood estimator. We develop the asymptotic theory of the estimators when the time horizon of the observations goes to infinity, considering both cases of a fixed time lag (low frequency) and a vanishing time lag (high frequency) between consecutive observations. In the setting of low frequency observations and infinite time horizon we also study the convergence of three local time estimators, that are already known to converge to the local time in the setting of high frequency observations and fixed time horizon. We find that these estimators can behave differently, depending on the assumptions on the time lag between observations.

math.ST↗
arXiv2023

Maximum likelihood estimator for skew Brownian motion: the convergence rate

We give a thorough description of the asymptotic property of the maximum likelihood estimator (MLE) of the skewness parameter of a Skew Brownian Motion (SBM). Thanks to recent results on the Central Limit Theorem of the rate of convergence of estimators for the SBM, we prove a conjecture left open that the MLE has asymptotically a mixed normal distribution involving the local time with a rate of convergence of order $1/4$. We also give a series expansion of the MLE and study the asymptotic behavior of the score and its derivatives, as well as their variation with the skewness parameter. In particular, we exhibit a specific behavior when the SBM is actually a Brownian motion, and quantify the explosion of the coefficients of the expansion when the skewness parameter is close to $-1$ or $1$.

math.ST↗
arXiv2022

Drift estimation of the threshold Ornstein-Uhlenbeck process from continuous and discrete observations

We refer by threshold Ornstein-Uhlenbeck to a continuous-time threshold autoregressive process. It follows the Ornstein-Uhlenbeck dynamics when above or below a fixed level, yet at this level (threshold) its coefficients can be discontinuous. We discuss (quasi)-maximum likelihood estimation of the drift parameters, both assuming continuous and discrete time observations. In the ergodic case, we derive consistency and speed of convergence of these estimators in long time and high frequency. Based on these results, we develop a test for the presence of a threshold in the dynamics. Finally, we apply these statistical tools to short-term US interest rates modeling.

math.PR↗
arXiv2022

On moments and strong local Hölder regularity of solutions of stochastic differential equations and of their spatial derivative processes

Spatial differentiability of solutions of stochastic differential equations (SDEs) is a classical question in stochastic analysis. The case of coefficients with globally Lipschitz continuous derivatives is well understood in the literature. Counterexamples with smooth and bounded coefficients demonstrate that the non-globally Lipschitz case is more subtle. In this article we establish conditions, including a suitable local monotonicity property, which provide existence of continuously differentiable solutions of SDEs, moment estimates and strong local Hölder regularity.

math.PR↗
arXiv2019

A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations

There are numerous applications of the classical (deterministic) Gronwall inequality. Recently, Michael Scheutzow discovered a stochastic Gronwall inequality which provides upper bounds for $p$-th moments, $p\in(0,1)$, of the supremum of nonnegative scalar continuous processes which satisfy a linear integral inequality. In this article we complement this with upper bounds for $p$-th moments, $p\in[2,\infty)$, of the supremum of general Itô processes which satisfy a suitable one-sided affine-linear growth condition. As example applications, we improve known results on strong local Lipschitz continuity in the starting point of solutions of stochastic differential equations (SDEs), on (exponential) moment estimates for SDEs, on strong completeness of SDEs, and on perturbation estimates for SDEs.

math.PR↗
arXiv2019

Existence, uniqueness, and numerical approximations for stochastic Burgers equations

In this paper we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existent fully explicit space-time discrete approximation scheme and, in particular, the fact that it satisfies suitable a priori estimates. As a byproduct we obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the paper to the stochastic Burgers equations with space-time white noise.

math.PR↗
arXiv2018

Existence and uniqueness properties for solutions of a class of Banach space valued evolution equations

In this note we provide a self-contained proof of an existence and uniqueness result for a class of Banach space valued evolution equations with an additive forcing term. The framework of our abstract result includes, for example, finite dimensional ordinary differential equations (ODEs), semilinear deterministic partial differential equations (PDEs), as well as certain additive noise driven stochastic partial differential equations (SPDEs) as special cases. The framework of our general result assumes somehow mild regularity conditions on the involved semigroup and also allows the involved semigroup operators to be nonlinear. The techniques used in the proofs of our results are essentially well-known in the relevant literature. The contribution of this note is to provide a rather general existence and uniqueness result which covers several situations as special cases and also to provide a self-contained proof for this existence and uniqueness result.

math.CA↗
arXiv2018

Strong convergence for explicit space-time discrete numerical approximation for 2D stochastic Navier-Stokes equations

In this paper we show the strong convergence of a fully explicit space-time discrete approximation scheme for the solution process of the two-dimensional incompressible stochastic Navier-Stokes equations on the torus driven by additive noise. To do so we apply an existing result which was designed to prove strong convergence for the same approximation method for other stochastic partial differential equations with non-globally monotone non-linearities.

math.PR↗
arXiv2016

Exact simulation of Brownian diffusions with drift admitting jumps

In this paper, using an algorithm based on the retrospective rejection sampling scheme, we propose an exact simulation of a Brownian diffusion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical diffculty due to the presence of two jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift.

math.PR↗