arXiv subjects

Giacomo Ascione

Publications and source records attributed to Giacomo Ascione.

At least 19 recordsLinked to original sources

arXiv2026

Time-changed Markov processes and space-time coupled non-local equations

In this paper we study coupled fully non-local equations, where a linear non-local operator jointly acts on the time and space variables. We establish existence and uniqueness of the solution. A maximum principle is proved and used to derive uniqueness. Existence is established by providing a stochastic representation based on anomalous processes constructed as a time change via the undershooting of an independent subordinator. This leads to general non-stepped processes with intervals of constancy representing a sticky or trapping effect. Our theory allows these intervals to be dependent on the immediately subsequent jump. These processes include scaling limit of suitable coupled continuous time random walks previously studied in applications, in particular in the context of anomalous diffusion and option pricing. Here we exploit our general theory to obtain a non-local analog of the Black and Scholes equation, addressing the problem of determining the seasoned price of a derivative security, in case the price fluctuations are described by a process whose jumps are dependent on the previous interval.

math.PR↗

arXiv2026

Non-local heat equations with moving boundary

In this paper we consider non-local (in time) heat equations on time-increasing parabolic sets whose boundary is determined by a suitable curve. We provide a notion of solution for these equations and we study well-posedness under Dirichlet conditions outside the domain. A maximum principle is proved and used to derive uniqueness and continuity with respect to the initial datum of the solutions of the Dirichlet problem. Existence is proved by showing a stochastic representation based on the delayed Brownian motion killed on the boundary. Several related distributional properties of the delayed Brownian motion and its crossing probabilities are also obtained. The asymptotic behaviour of the mean square displacement of the process is determined, showing that the diffusive behaviour is anomalous.

math.PR↗

arXiv2026

Scaling limits for some Mittag-Leffler queues

In this paper, we consider five models of heavy-tailed queues involving Mittag-Leffler distributions that generalize the classical $M/M/1$ queues. These models are suitable modifications of previously defined models in such a way that the classical $M/M/1$ queue can be recovered by a suitable selection of parameters. We provide the distribution of inter-arrival and service times of both the original and modified queueing models. We then study the scaling limits of all the proposed models and we argue that the behaviour of the limiting processes can be used to characterise the traffic regime of the queues.

math.PR↗

arXiv2025

Special potentials for relativistic Laplacians I: Fractional Rollnik-class

We propose a counterpart of the classical Rollnik-class of potentials for fractional and massive relativistic Laplacians, and describe this space in terms of appropriate Riesz potentials. These definitions rely on precise resolvent estimates. We show that Coulomb-type potentials are elements of fractional Rollnik-class up to but not including the critical singularity of the Hardy potential. For the operators with fractional exponent $α= 1$ there exists no fractional Rollnik potential, however, in low dimensions we make sense of these classes as limiting cases by using $Γ$-convergence. In a second part of the paper we derive detailed results on the self-adjointness and spectral properties of relativistic Schrödinger operators obtained under perturbations by fractional Rollnik potentials. We also define an extended fractional Rollnik-class which is the maximal space for the Hilbert-Schmidt property of the related Birman-Schwinger operators.

math.FA↗

arXiv2024

Optimal control problems driven by nonlinear degenerate Fokker-Planck equations

The well-posedness of a class of optimal control problems is analysed, where the state equation couples a nonlinear degenerate Fokker-Planck equation with a system of Ordinary Differential Equations (ODEs). Such problems naturally arise as mean-field limits of Stochastic Differential models for multipopulation dynamics, where a large number of agents (followers) is steered through parsimonious intervention on a selected class of leaders. The proposed approach combines stability estimates for measure solutions of nonlinear degenerate Fokker-Planck equations with a general framework of assumptions on the cost functional, ensuring compactness and lower semicontinuity properties. The Lie structure of the state equations allows one for considering non-Lipschitz nonlinearities, provided some suitable dissipativity assumptions are considered in addition to non-Euclidean Hölder and sublinearity conditions.

math.OC↗

arXiv2024

Well-posedness of Kolmogorov-Fokker-Planck equations with unbounded drift

We consider Kolmogorov-Fokker-Planck equations with unbounded drift terms which are only measurable in time and locally Hölder continuous in space. In particular, we extend the parametrix method to this setting and we prove existence and uniqueness of measure solutions to the associated Cauchy problem, as well as the equivalence with the corresponding stochastic formulation.

math.AP↗

arXiv2024

On the ergodicity of a three-factor CIR model

This study introduces the CIR3 model, a three-factor model characterized by stochastic and correlated trends and volatilities. The paper focuses on establishing the Wasserstein ergodicity of this model, a task not achievable through conventional means such as the Dobrushin theorem. Instead, alternative mathematical approaches are employed, including considerations of topological aspects of Wasserstein spaces and Kolmogorov equations for measures. Remarkably, the methodology developed here can also be applied to prove the Wasserstein ergodicity of the widely recognized three-factor Chen model.

math.PR↗

arXiv2024

Time changed spherical Brownian motions with longitudinal drifts

In this paper, we consider the time change of the diffusion process on the 2-dimensional unit sphere generated by the Laplace-Beltrami operator, perturbed by means of a longitudinal vector field. First, this is done by addressing the problem of finding strong solutions to suitable time-nonlocal Kolmogorov equations, via a spectral decomposition approach. Next, the desired process is constructed as the composition of the aforementioned diffusion process and the inverse of a subordinator, and it is used to provide a stochastic representation of the solution of the involved time-nonlocal Kolmogorov equation, which in turn leads to the spectral decomposition of its probability density function. A family of operators induced by the process is then adopted to provide very weak solutions of the same time-nonlocal Kolmogorov equation with much less regular initial data. From the spectral decomposition results we also get some bounds on the speed of convergence to the stationary state, proving that the process can be considered an anomalous diffusion. These results improve some known ones in terms of both the presence of a perturbation and the lack of regularity of the initial data.

math.PR↗

arXiv2024

Regularity and asymptotics of densities of inverse subordinators

In this article densities (and their derivatives) of subordinators and inverse subordinators are considered. Under minor restrictions, generally milder than the existing in the literature, using a useful modification of the saddle point method, we obtain the large asymptotic behaviour of these densities (and their derivatives) for a specific region of space and time and quantify how the ratio between time and space affects the explicit speed of convergence. The asymptotics is governed by an exponential term depending on the Laplace exponent of the subordinator and the region represents the behaviour of the subordinator when it is atypically small (the inverse one is larger than usual). As a result a route to the derivation of novel general or particular fine estimates for densities with explicit constants in the speed of convergence in the region of the lower envelope/the law of iterated logarithm is available. Furthermore, under mild conditions, we provide a power series representation for densities (and their derivatives) of subordinators and inverse subordinators. This representation is explicit and based on the derivatives of the convolution of the tails of the corresponding Lévy measure, whose smoothness is also investigated. In this context the methods adopted are based on Laplace inversion and strongly rely on the theory of Bernstein functions extended to the cut complex plane. As a result, smoothness properties of densities (and their derivatives) and their behaviour near zero immediately follow.

math.PR↗

arXiv2023

Modeling Volatility of Disaster-Affected Populations: A Non-Homogeneous Geometric-Skew Brownian Motion Approach

This paper delves into the impact of natural disasters on affected populations and underscores the imperative of reducing disaster-related fatalities through proactive strategies. On average, approximately 45,000 individuals succumb annually to natural disasters amid a surge in economic losses. The paper explores catastrophe models for loss projection, emphasizes the necessity of evaluating volatility in disaster risk, and introduces an innovative model that integrates historical data, addresses data skewness, and accommodates temporal dependencies to forecast shifts in mortality. To this end, we introduce a time-varying skew Brownian motion model, for which we provide proof of the solution's existence and uniqueness. In this model, parameters change over time, and past occurrences are integrated via volatility.

stat.AP↗

arXiv2022

Bulk behaviour of ground states for relativistic Schrödinger operators with compactly supported potentials

We propose a probabilistic representation of the ground states of massive and massless Schrödinger operators with a potential well in which the behaviour inside the well is described in terms of the moment generating function of the first exit time from the well, and the outside behaviour in terms of the Laplace transform of the first entrance time into the well. This allows an analysis of their behaviour at short to mid-range from the origin. In a first part we derive precise estimates on these two functionals for stable and relativistic stable processes. Next, by combining scaling properties and heat kernel estimates, we derive explicit local rates of the ground states of the given family of non-local Schrödinger operators both inside and outside the well. We also show how this approach extends to fully supported decaying potentials. By an analysis close-by to the edge of the potential well, we furthermore show that the ground state changes regularity, which depends qualitatively on the fractional power of the non-local operator.

math.SP↗

arXiv2022

Stability of ground state eigenvalues of non-local Schrödinger operators with respect to potentials and applications

In a first part of this paper we investigate the continuity (stability) of the spectrum of a class of non-local Schrödinger operators on varying the potentials. By imposing conditions of different strength on the convergence of the sequence of potentials, we give either direct proofs to show the strong or norm resolvent convergence of the so-obtained sequence of non-local Schrödinger operators, or via $Γ$-convergence of the related positive forms for more rough potentials. In a second part we use these results to show via a sequence of suitably constructed approximants that the ground states of massive or massless relativistic Schrödinger operators with spherical potential wells are radially decreasing functions.

math.AP↗

arXiv2022

Foreign exchange options on Heston-CIR model under Lévy process framework

In this paper, we consider the Heston-CIR model with Lévy process for pricing in the foreign exchange (FX) market by providing a new formula that better fits the distribution of prices. To do that, first, we study the existence and uniqueness of the solution to this model. Second, we examine the strong convergence of the Lévy process with stochastic domestic short interest rates, foreign short interest rates and stochastic volatility. Then, we apply Least Squares Monte Carlo (LSM) method for pricing American options under our model with stochastic volatility and stochastic interest rate. Finally, by considering real-world market data, we illustrate numerical results for the four-factor Heston-CIR Lévy model.

math.PR↗

arXiv2022

A sojourn-based approach to semi-Markov Reinforcement Learning

In this paper we introduce a new approach to discrete-time semi-Markov decision processes based on the sojourn time process. Different characterizations of discrete-time semi-Markov processes are exploited and decision processes are constructed by their means. With this new approach, the agent is allowed to consider different actions depending also on the sojourn time of the process in the current state. A numerical method based on $Q$-learning algorithms for finite horizon reinforcement learning and stochastic recursive relations is investigated. Finally, we consider two toy examples: one in which the reward depends on the sojourn-time, according to the gambler's fallacy; the other in which the environment is semi-Markov even if the reward function does not depend on the sojourn time. These are used to carry on some numerical evaluations on the previously presented $Q$-learning algorithm and on a different naive method based on deep reinforcement learning.

math.PR↗

arXiv2022

Mean field sparse optimal control of systems with additive white noise

We analyze the problem of controlling a multi-agent system with additive white noise through parsimonious interventions on a selected subset of the agents (leaders). For such a controlled system with a SDE constraint, we introduce a rigorous limit process towards an infinite dimensional optimal control problem constrained by the coupling of a system of ODE for the leaders with a McKean-Vlasov-type SDE, governing the dynamics of the prototypical follower. The latter is, under some assumptions on the distribution of the initial data, equivalent with a (nonlinear parabolic) PDE-ODE system. The derivation of the limit mean-field optimal control problem is achieved by linking the mean-field limit of the governing equations together with the $Γ$-limit of the cost functionals for the finite dimensional problems.

math.OC↗

arXiv2021

A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem

We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is $1/2$ and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.

math.FA↗

arXiv2021

Deterministic control of SDEs with stochastic drift and multiplicative noise: a variational approach

We consider a linear stochastic differential equation with stochastic drift and multiplicative noise. We study the problem of approximating its solution with the process that solves the equation where the possibly stochastic drift is replaced by a deterministic function. To do this, we use a combination of deterministic Pontryagin's maximum principle approach and direct methods of calculus of variations. We find necessary and sufficient conditions for a function $u \in L^1(0,T)$ to be a minimizer of a certain cost functional. To overcome the problem of the existence of such minimizer, we also consider suitable families of penalized coercive cost functionals. Finally, we consider the important example of the quadratic cost functional, showing that the expected value of the drift component is not always the best choice in the mean squared error approximation.

math.PR↗

arXiv2021

A matrix Harnack inequality for semilinear heat equations

We derive a matrix version of Li \& Yau--type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R.~Hamilton did in~\cite{hamilton7} for the standard heat equation. We then apply these estimates to obtain some Harnack--type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.

math.AP↗