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Stefano Bonaccorsi

Publications and source records attributed to Stefano Bonaccorsi.

At least 19 recordsLinked to original sources
arXiv2025

Limit theorems for stochastic Volterra processes

We introduce an abstract Hilbert space-valued framework of Markovian lifts for stochastic Volterra equations with operator-valued Volterra kernels. Our main results address the existence and characterisation of possibly multiple limit distributions and stationary processes, a law of large numbers including a convergence rate, and the central limit theorem for time averages of the process within the Gaussian domain of attraction. As particular examples, we study Markovian lifts based on Laplace transforms in a weighted Hilbert space of densities and Markovian lifts based on the shift semigroup on the Filipovi\'c space. We illustrate our results for the case of fractional stochastic Volterra equations with additive or multiplicative Gaussian noise.

math.PR↗
arXiv2024

Markovian lifting and optimal control for integral stochastic Volterra equations with completely monotone kernels

In this paper, we focus on solving the optimal control problem for integral stochastic Volterra equations in a finite dimensional setting. In our setting, the noise term is driven by a pure jump L\'evy noise and the control acts on the intensity of the jumps. We use recent techniques proposed by Hamaguchi, where a crucial requirement is that the convolution kernel should be a completely monotone function. This allows us to use Bernstein's representation and the machinery of Laplace transform to obtain a Markovian lift. It is natural that the Markovian lift, in whatever form constructed, transforms the state equation into a stochastic differential equation in an infinite-dimensional space. This space should be large enough to contain all the information about the history of the process. Hence, although the original equation is taken in a finite dimensional space, the resulting lift is always infinite dimensional. We solve the problem by using the forward-backward approach in the infinite-dimensional setting and prove the existence of the optimal control for the original problem. Under additional assumptions on the coefficients, we see that a control in closed-loop form can be achieved.

math.OC↗
arXiv2023

Limits of stochastic Volterra equations driven by Gaussian noise

We study stochastic Volterra equations in Hilbert spaces driven by cylindrical Gaussian noise. We derive a mild formulation for the stochastic Volterra equation, prove the equivalence of mild and strong solutions, the existence and uniqueness of mild solutions, and study space-time regularity. Furthermore, we establish the stability of mild solutions in $L^q(\R_+)$, prove the existence of limit distributions in the Wasserstein $p$-distance with $p \in [1,\infty)$, and characterise when these limit distributions are independent of the initial state of the process despite the presence of memory. While our techniques allow for a general class of Volterra kernels, they are particularly suited for completely monotone kernels and fractional Riemann-Liouville kernels in the full range $α\in (0,2)$.

math.PR↗
arXiv2023

Sticky Brownian motions on star graphs

This paper is concerned with the construction of several stochastic processes in a star graph, that is a non-euclidean structure where some features of the classical modelling fail. We propose a model for trapping phenomena with characterization of the traps in terms of a singular measure. This measure also defines a non-local operator by means of which we introduce a non-local dynamic condition for the parabolic problem on the star graph. We study semi-Markov processes on the rays of the graph in order to obtain a probabilistic representation of the motion on the whole graph. Extensions to general graph structures can be given by applying our results on star graphs.

math.PR↗
arXiv2023

Non-negativity and zero isolation for generalized mixtures of densities

In the literature, finite mixture models are described as linear combinations of probability distribution functions having the form $\displaystyle f(x) = Λ\sum_{i=1}^n w_i f_i(x)$, $x \in \mathbb{R}$, where $w_i$ are positive weights, $Λ$ is a suitable normalising constant and $f_i(x)$ are given probability density functions. The fact that $f(x)$ is a probability density function follows naturally in this setting. Our question is: what happens when we remove the sign condition on the coefficients $w_i$? The answer is that it is possible to determine the sign pattern of the function $f(x)$ by an algorithm based on finite sequence that we call a generalized Budan-Fourier sequence. In this paper we provide theoretical motivation for the functioning of the algorithm, and we describe with various examples its strength and possible applications.

math.ST↗
arXiv2022

A stochastic differential equation SIS model on network under Markovian switching

We study a stochastic SIS (susceptible-infected-susceptible) epidemic dynamics on network, under the effect of a Markovian regime-switching. We first prove the existence of a unique global positive solution, and find a positive invariant set for the system. Then, we find sufficient conditions for a.s. extinction and stochastic permanence, showing also their relation with the stationary probability distribution of the Markov chain that governs the switching and with the network topology. We provide an asymptotic lower bound for the time average of the sample-path solution under the conditions ensuring stochastic permanence. From this bound, we are able to prove the existence of an invariant probability measure if the condition of stochastic permanence holds. Under a different condition, we prove the positive recurrence and the ergodicity of the regime-switching diffusion.

math.PR↗
arXiv2022

A class of fractional Ornstein-Uhlenbeck processes mixed with a Gamma distribution

We consider a sequence of fractional Ornstein-Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.

math.PR↗
arXiv2022

Non-local Boundary Value Problems, stochastic resetting and Brownian motions on graphs

We consider dynamic boundary conditions involving non-local operators. Our analysis includes a detailed description of such operators together with their relations with random times and random (additive) functionals. We provide some new characterizations for the boundary behaviour of the Brownian motion based on the interplay between non-local operators and boundary value problems. Our main focus is on Feller-Wentzell diffusions with jumps (resetting/restart). We first consider the instructive case of the real line, then we extend our results on star graphs with trapping points or repulsive vertices.

math.PR↗
arXiv2020

Some aspects of the Markovian SIRS epidemic on networks and its mean-field approximation

We study the spread of an SIRS-type epidemic with vaccination on network. Starting from an exact Markov description of the model, we investigate the mean epidemic lifetime by providing a sufficient condition for fast extinction that depends on the model parameters and the topology of the network. Then, we pass to consider a firstorder mean-field approximation of the exact model and its stability properties, by relying on the graph-theoretical notion of equitable partition. In the case of graphs possessing this kind of partition, we prove that the endemic equilibrium can be computed by using a lower-dimensional dynamical system. Finally, in the special case of regular graphs, we investigate the domain of attraction of the endemic equilibrium.

math.DS↗
arXiv2020

Random evolution equations: well-posedness, asymptotics, and applications to graphs

We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for pathwise convergence in norm of the (random) propagator towards a (deterministic) steady state. We apply our findings in two environments with randomly evolving features: ensembles of difference operators on combinatorial graphs, or else of differential operators on metric graphs.

math.DS↗
arXiv2018

Surface measures and integration by parts formula on levels sets induced by functionals of the Brownian motion in $\mathbb R^n$

On the infinite dimensional space $E$ of continuous paths from $[0,1]$ to $\mathbb R^n$, $n \ge 3$, endowed with the Wiener measure $μ$, we construct a surface measure defined on level sets of the $L^2$-norm of $n$-dimensional processes that are solutions to a class of stochastic gradient system-type equations, and provide an integration by parts formula involving this surface measure. We follow the approach to surface measures in Gaussian spaces proposed via techniques of Malliavin calculus by Airault and Malliavin in 1988.

math.PR↗
arXiv2017

Optimal curing policy for epidemic spreading over a community network with heterogeneous population

The design of an efficient curing policy, able to stem an epidemic process at an affordable cost, has to account for the structure of the population contact network supporting the contagious process. Thus, we tackle the problem of allocating recovery resources among the population, at the lowest cost possible to prevent the epidemic from persisting indefinitely in the network. Specifically, we analyze a susceptible-infected-susceptible epidemic process spreading over a weighted graph, by means of a first-order mean-field approximation. First, we describe the influence of the contact network on the dynamics of the epidemics among a heterogeneous population, that is possibly divided into communities. For the case of a community network, our investigation relies on the graph-theoretical notion of equitable partition; we show that the epidemic threshold, a key measure of the network robustness against epidemic spreading, can be determined using a lower-dimensional dynamical system. Exploiting the computation of the epidemic threshold, we determine a cost-optimal curing policy by solving a convex minimization problem, which possesses a reduced dimension in the case of a community network. Lastly, we consider a two-level optimal curing problem, for which an algorithm is designed with a polynomial time complexity in the network size.

math.OC↗
arXiv2017

Probabilistic representation formula for the solution of fractional high order heat-type equations

We propose a probabilistic construction for the solution of a general class of fractional high order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time change governed by a class of subordinated processes allows to handle the fractional part of the derivative in space. We first consider evolution equations with space fractional derivatives of any order, and later we show the extension to equations with time fractional derivative (in the sense of Caputo derivative) of order $α\in (0,1)$.

math.PR↗
arXiv2017

Maximum principle for an optimal control problem associated to a SPDE with nonlinear boundary conditions

We study a control problem where the state equation is a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise. We allow the control to act on the boundary and set stochastic boundary conditions that depend on the time derivative of the solution on the boundary. This work provides necessary and sufficient conditions of optimality in the form of a maximum principle. We also provide a result of existence for the optimal control in the case where the control acts linearly.

math.PR↗
arXiv2016

Construction of a surface integral under local Malliavin assumption and integration by parts formulae

In this paper, we consider convex sets $K_r = \{g \ge r\}$ in an infinite dimensional Hilbert space, where $g$ is suitably related to a reference Gaussian measure $μ$ in $H$. We first show how to define a surface measure on the level sets $\{g = r\}$ that is related to $μ$. This allows to introduce an integration-by-parts formula in $H$. This formula can be applied in several important constructions, as for instance the case where $μ$ is the law of a (Gaussian) stochastic process and $H$ is the space of its trajectories

math.PR↗
arXiv2016

Epidemics on networks with heterogeneous population and stochastic infection rates

In this paper we study the diffusion of an SIS-type epidemics on a network under the presence of a random environment, that enters in the definition of the infection rates of the nodes. Accordingly, we model the infection rates in the form of independent stochastic processes. To analyze the problem, we apply a mean field approximation, which allows to get a stochastic differential equations for the probability of infection in each node, and classical tools about stability, which require to find suitable Lyapunov's functions. Here, we find conditions which guarantee, respectively, extinction and stochastic persistence of the epidemics. We show that there exists two regions, given in terms of the coefficients of the model, one where the system goes to extinction almost surely, and the other where it is stochastic permanent. These two regions are, unfortunately, not adjacent, as there is a gap between them, whose extension depends on the specific level of noise. In this last region, we perform numerical analysis to suggest the true behavior of the solution.

math.PR↗
arXiv2016

Absolute continuity of the law for solutions of stochastic differential equations with boundary noise

We study existence and regularity of the density for the solution $u(t,x)$ (with fixed $t > 0$ and $x \in D$) of the heat equation in a bounded domain $D \subset \mathbb R^d$ driven by a stochastic inhomogeneous Neumann boundary condition with stochastic term. The stochastic perturbation is given by a fractional Brownian motion process. Under suitable regularity assumptions on the coefficients, by means of tools from the Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure in $\mathbb R$.

math.PR↗