arXiv subjects

Alberto Lanconelli

Publications and source records attributed to Alberto Lanconelli.

At least 19 recordsLinked to original sources

arXiv2026

The Tricomi equation in the hyperbolic half plane under additive space-time Gaussian White Noise perturbation

We study the Cauchy problem for the Tricomi equation perturbed by space-time Gaussian White Noise. To prove existence and uniqueness of the solution, we employ a Fourier transform approach that allows to obtain its representation in terms of certain integrals of the Airy functions. Then, via a careful analysis of the asymptotic behaviour of those integrals, we obtain all the desired properties of the solution, such as square integrability, continuity of its sample paths and stationarity with respect to the space variable. In relation to that stationarity, we also provide the precise description of how the correlation function behaves for small values of the space-lag. We also remark that, in contrast to the findings of the recent paper [11], the properties of the solution to our stochastic Tricomi equation are equivalent to those derived in studying the corresponding problem for the wave operator.

math.PR↗

arXiv2026

Regularization by noise for Gevrey well-posedeness of a weakly hyperbolic operator

We present an example of a linear partial differential equation whose Cauchy problem becomes well-posed when perturbed by noise. Specifically, we make clear how a suitable multiplicative Stratonovich perturbation of Brownian type renders a weakly hyperbolic operator with double involutive characteristics well-posed in the $C^{\infty}$-category, while its deterministic counterpart is only well-posed in the Gevrey $ s $ classes with $ 1 \leq s <2 $ .

math.PR↗

arXiv2025

A note on Tricomi-type partial differential equations with white noise initial condition

We study a class of Tricomi-type partial differential equations previously investigated in [28]. Firstly, we generalize the representation formula for the solution obtained there by allowing the coefficient in front of the second-order partial derivative with respect to $x$ to be any non integer power of $t$. Then, we analyze the robustness of that solution by taking the initial data to be Gaussian white noise and we discover that the existence of a well-defined random field solution is lost upon the introduction of lower-order terms in the operator. This phenomenon shows that, even though the Tricomi-type operators with or without lower-order terms are the same from the point of view of the theory of hyperbolic operators with double characteristics, their corresponding random versions exhibit different well posedness properties. We also prove that for more regular initial data, specifically fractional Gaussian white noise with Hurst parameter $H\in (1/2,1)$, the well posedness of the Cauchy problem for the Tricomi-type operator with lower-order term is restored.

math.PR↗

arXiv2024

A new numerical scheme for It\^o stochastic differential equations based on Wick-type Wong-Zakai arguments

The aim of this note is to propose a novel numerical scheme for drift-less one dimensional stochastic differential equations of It\^o's type driven by standard Brownian motion. Our approximation method is equivalent to the well known Milstein scheme as long as the rate of convergence is concerned, i.e. it is strongly convergent with order one, but has the additional desirable property of being exact for linear diffusion coefficients. Our approach is inspired by Wick-type Wong-Zakai arguments in the sense that we only smooth the white noise through polygonal approximation of the Brownian motion while keep the equation in differential form. A first order Taylor expansion of the diffusion coefficient allows us to solve the resulting equation explicitly and hence to provide an implementable approximation scheme.

math.PR↗

arXiv2024

Non trivial optimal sampling rate for estimating a Lipschitz-continuous function in presence of mean-reverting Ornstein-Uhlenbeck noise

We examine a mean-reverting Ornstein-Uhlenbeck process that perturbs an unknown Lipschitz-continuous drift and aim to estimate the drift's value at a predetermined time horizon by sampling the path of the process. Due to the time varying nature of the drift we propose an estimation procedure that involves an online, time-varying optimization scheme implemented using a stochastic gradient ascent algorithm to maximize the log-likelihood of our observations. The objective of the paper is to investigate the optimal sample size/rate for achieving the minimum mean square distance between our estimator and the true value of the drift. In this setting we uncover a trade-off between the correlation of the observations, which increases with the sample size, and the dynamic nature of the unknown drift, which is weakened by increasing the frequency of observation. The mean square error is shown to be non monotonic in the sample size, attaining a global minimum whose precise description depends on the parameters that govern the model. In the static case, i.e. when the unknown drift is constant, our method outperforms the arithmetic mean of the observations in highly correlated regimes, despite the latter being a natural candidate estimator. We then compare our online estimator with the global maximum likelihood estimator.

math.ST↗

arXiv2024

A Novel Theoretical Framework for Exponential Smoothing

Simple Exponential Smoothing is a classical technique used for smoothing time series data by assigning exponentially decreasing weights to past observations through a recursive equation; it is sometimes presented as a rule of thumb procedure. We introduce a novel theoretical perspective where the recursive equation that defines simple exponential smoothing occurs naturally as a stochastic gradient ascent scheme to optimize a sequence of Gaussian log-likelihood functions. Under this lens of analysis, our main theorem shows that -- in a general setting -- simple exponential smoothing converges to a neighborhood of the trend of a trend-stationary stochastic process. This offers a novel theoretical assurance that the exponential smoothing procedure yields reliable estimators of the underlying trend shedding light on long-standing observations in the literature regarding the robustness of simple exponential smoothing.

stat.ME↗

arXiv2024

A new look to branching Brownian motion from a particle based reaction diffusion dynamics point of view

Aim of this note is to analyse branching Brownian motion within the class of models introduced in the recent paper [4] and called chemical diffusion master equations. These models provide a description for the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles. We derive an infinite system of Fokker-Planck equations that rules the probabilistic evolution of the single particles generated by the branching mechanism and analyse its properties using Malliavin Calculus techniques, following the ideas proposed in [13]. Another key ingredient of our approach is the McKean representation for the solution of the Fisher-Kolmogorov-Petrovskii-Piskunov equation and a stochastic counterpart of that equation. We also derive the reaction-diffusion partial differential equation solved by the average concentration field of the branching Brownian system of particles.

math.PR↗

arXiv2023

Probabilistic derivation and analysis of the chemical diffusion master equation with mutual annihilation

We propose a probabilistic derivation of the so-called chemical diffusion master equation (CDME) and describe an infinite dimensional moment generating function method for finding its analytical solution. CDMEs model by means of an infinite system of coupled Fokker-Planck equations the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles; here, we focus an creation and mutual annihilation chemical reactions combined with Brownian diffusion of the single particles. Our probabilistic approach mimics the derivation of backward Kolmogorov equations for birth-death continuous time Markov chains. Moreover, the proposed infinite dimensional moment generating function method links certain finite dimensional projections of the solution of the CDME to the solution of a single linear fourth order partial differential equation containing as many variables as the dimension of the aforementioned projection space.

math.PR↗

arXiv2023

Maximum Likelihood With a Time Varying Parameter

We consider the problem of tracking an unknown time varying parameter that characterizes the probabilistic evolution of a sequence of independent observations. To this aim, we propose a stochastic gradient descent-based recursive scheme in which the log-likelihood of the observations acts as time varying gain function. We prove convergence in mean-square error in a suitable neighbourhood of the unknown time varying parameter and illustrate the details of our findings in the case where data are generated from distributions belonging to the exponential family.

math.ST↗

arXiv2023

Solution formula for the general birth-death chemical diffusion master equation

We propose a solution formula for chemical diffusion master equations of birth and death type. These equations, proposed and formalized in the recent paper [5], aim at incorporating the spatial diffusion of molecules into the description provided by the classical chemical master equation. We start from the general approach developed in [20] and perform a more detailed analysis of the representation found there. This leads to a solution formula for birth-death chemical diffusion master equations which is expressed in terms of the solution to the reaction-diffusion partial differential equation associated with the system under investigation. Such representation also reveals a striking analogy with the solution to the classical birth-death chemical master equations. The solutions of our findings are also illustrated for several examples.

math.PR↗

arXiv2022

A note on diffusion limits for stochastic gradient descent

In the machine learning literature stochastic gradient descent has recently been widely discussed for its purported implicit regularization properties. Much of the theory, that attempts to clarify the role of noise in stochastic gradient algorithms, has widely approximated stochastic gradient descent by a stochastic differential equation with Gaussian noise. We provide a novel rigorous theoretical justification for this practice that showcases how the Gaussianity of the noise arises naturally.

cs.LG↗

arXiv2022

Using Malliavin calculus to solve a chemical diffusion master equation

We propose a novel method to solve a chemical diffusion master equation of birth and death type. This is an infinite system of Fokker-Planck equations where the different components are coupled by reaction dynamics similar in form to a chemical master equation. This system was proposed in [3] for modelling the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles. Using some basic tools and ideas from infinite dimensional Gaussian analysis we are able to reformulate the aforementioned infinite system of Fokker-Planck equations as a single evolution equation solved by a generalized stochastic process and written in terms of Malliavin derivatives and differential second quantization operators. Via this alternative representation we link certain finite dimensional projections of the solution of the original problem to the solution of a single partial differential equations of Ornstein-Uhlenbeck type containing as many variables as the dimension of the aforementioned projection space.

math.PR↗

arXiv2021

On a class of stochastic hyperbolic equations with double characteristics

We study the effect of Gaussian perturbations on a hyperbolic partial differential equation with double characteristics in two spatial dimensions. The coefficients of our partial differential operator depend polynomially on the space variables, while the noise is additive, white in time and coloured in space. We provide a sufficient condition on the spectral measure of the covariance functional describing the noise that allows for the existence of a random field solution for the resulting stochastic partial differential equation. Our approach is based on explicit computations for the fundamental solution of the partial differential operator and its Fourier transform.

math.PR↗

arXiv2021

A small time approximation for the solution to the Zakai Equation

We propose a novel small time approximation for the solution to the Zakai equation from nonlinear filtering theory. We prove that the unnormalized filtering density is well described over short time intervals by the solution of a deterministic partial differential equation of Kolmogorov type; the observation process appears in a pathwise manner through the degenerate component of the Kolmogorov's type operator. The rate of convergence of the approximation is of order one in the lenght of the interval. Our approach combines ideas from Wong-Zakai-type results and Wiener chaos approximations for the solution to the Zakai equation. The proof of our main theorem relies on the well-known Feynman-Kac representation for the unnormalized filtering density and careful estimates which lead to completely explicit bounds.

math.PR↗

arXiv2021

On a new method for the stochastic perturbation of the disease transmission coefficient in SIS Models

In this study we investigate a novel approach to stochastically perturb the disease transmission coefficient, which is a key parameter in susceptible-infected-susceptible (SIS) models. Motivated by the papers [2] and [5], we perturb the disease transmission coefficient with a Gaussian white noise, formally modelled as the time derivative of a mean reverting Ornstein-Uhlenbeck process. We remark that, thanks to a suitable representation of the solution to the deterministic SIS model, this perturbation is rigorous and supported by a Wong-Zakai approximation argument that consists in smoothing the singular Gaussian white noise and then taking limit of the solution from the approximated model. We prove that the stochastic version of the classic SIS model obtained this way preserves a crucial feature of the deterministic equation: the reproduction number dictating the two possible asymptotic regimes for the infection, i.e. extinction and persistence, remains unchanged. We then identify the class of perturbing noises for which this property holds and propose simple sufficient conditions for that. All the theoretical discoveries are illustrated and discussed with the help of several numerical simulations.

math.PR↗

arXiv2021

Upper and lower bounds for the solution of a stochastic prey-predator system with foraging arena scheme

We investigate some probabilistic aspects of the unique global strong solution of a two dimensional system of stochastic differential equations describing a prey-predator model perturbed by Gaussian noise. We first establish, for any fixed $t> 0$, almost sure upper and lower bounds for the components $X(t)$ and $Y(t)$ of the solution vector: these explicit estimates emphasize the interplay between the various parameters of the model and agree with the asymptotic results found in the literature. Then, standing on the aforementioned bounds, we derive upper and lower estimates for the joint moments and distribution function of $(X(t),Y(t))$. Our analysis is based on a careful use of comparison theorems for stochastic differential equations and exploits several peculiar features of the noise driving the equation.

math.PR↗

arXiv2021

Wong-Zakai approximations for quasilinear systems of Itô's type stochastic differential equations

We extend to the multidimensional case a Wong-Zakai-type theorem proved by Hu and Øksendal in [7] for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô's SDEs, we consider for any finite partition of the time interval $[0,T]$ a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally lead to the investigation of a semilinear hyperbolic system of partial differential equations that we utilize for constructing a solution of the Wong-Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker-Planck equation and that the sequence converges to the solution of the Itô equation, as the mesh of the partition tends to zero.

math.PR↗

arXiv2021

A note about the invariance of the basic reproduction number for stochastically perturbed SIS models

We try to justify rigorously, using a Wong-Zakai approximation argument, the susceptible-infected-susceptible (SIS) stochastic differential equation proposed in [2]. We discover that according to this approach the "right" stochastic model to be considered should be the Stratonovich version of the Itô equation analyzed in [2]. Surprisingly, this alternative model presents the following feature: the threshold value characterizing the two different asymptotic regimes of the solution coincides with the one describing the classical SIS deterministic equation.

math.PR↗