# Webinars

The UMI PRISMA Group promotes a cycle of online seminars on probability and mathematical statistics. The webinars will take place on the first working Monday of each month in the time slot 16:00-18:00 (CET) starting from November 2021. Further details on the scheduled seminars will be released. Everyone interested is warmly invited to attend the webinars. To join the webinars, click on the title.

### Next webinars:

### Past webinars:

##### Abstract:

Discrete random structures, such as random partitions and discrete random measures, have emerged as effective tools for Bayesian modeling and have fueled exciting advances in density estimation, clustering, prediction, feature allocation and survival analysis. The Dirichlet process (DP) has undoubtedly emerged as a reference model, mostly due to its analytical tractability. Nonetheless, the DP shares also some well-known limitations that have spurred a very lively area of research aiming at the proposal and the investigation of more general and flexible discrete nonparametric priors. The talk will provide a broad overview of such classes of priors and will specifically focus on those obtained as normalization of completely random measures. Characterizations of the induced random partitions and predictive rules will be illustrated and their role in designing computational algorithms for the approximation of Bayesian inferences of interest will be highlighted, both in exchangeable and non-exchangeable settings.

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The seminal work of Ferguson (1973), who introduced the Dirichlet process, has spurred the definition and investigation of more general classes of Bayesian nonparametric priors, with the aim at increasing flexibility while maintaining analytical tractability. Among the numerous generalizations, a fundamental class of random probability measures has been introduced by Regazzini et al. (2003): this is the class of normalized random measures with independent increments (NRMIs). NRMIs are random probability measures with almost surely discrete realizations, defined through the specifications of two ingredients: i) a sequence of unnormalized weights, which are the jumps of a Levy process on the positive real line; ii) a sequence of i.i.d. random atoms from a common base measure. The proposed construction is appealing from a mathematical standpoint, because analytical tractability is preserved, however NRMIs do not allow interaction among atoms, which are supposed to be independent and identically distributed. In some applied frameworks, the i.i.d. assumption could be too restrictive, for instance, in model-based clustering, when they are used as mixing measures in mixture models. To overcome this limitation, we propose a new class of normalized random measures with atoms' interaction. In our construction the atoms come from a finite point process, which is marked with i.i.d. positive weights. Thus, a new class of random probability measures is obtained by normalization. The desired interaction among atoms is then induced by a suitable choice of the law of the point process, which can create a repulsive or attractive behaviour. By means of Palm calculus, we are able to characterize marginal, predictive and posterior distributions for the proposed model. We specialize all our results for several choices of the finite point process, i.e., in the Determinantal, Gibbs and Shot-Noise Cox case. (Based on a joint work with Raffaele Argiento, Mario Beraha and Alessandra Guglielmi.)

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The emergence of periodic behavior in the dynamics of several interacting components is a common pattern of self-organisation in living systems. Rigorous mathematical treatments are still limited to few examples. In this talk we review some of these examples, emphasizing the essential role of the noise in the appearance of regular rhythms.

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In this talk we aim at establishing large deviation estimates for the probability that a simple random walk on the Euclidean lattice (d>2) covers a substantial fraction of a macroscopic body. It turns out that, when such rare event happens, the random walk is locally well approximated by random interlacements with a specific intensity, which can be used as a pivotal tool to obtain precise exponential rates. Random interlacements have been introduced by Sznitman in 2007 in order to describe the local picture left by the trace of a random walk on a large discrete torus when it runs up to times proportional to the volume of the torus, and has been since a popular object of study. In the first part of the talk we introduce random interlacements and give a brief account of some results surrounding this object. In the second part of the talk we study the event that random interlacements cover a substantial fraction of a macroscopic body. This allows to obtain an upper bound on the probability of the corresponding event for the random walk. Finally, by constructing a near-optimal strategy for the random walk to cover a macroscopic body, we discuss a matching large deviation lower bound. The talk is based on ongoing work with M. Nitzschner (NYU Courant).

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Models of neurons aim to describe the information transmission within a neural network. Some models are mathematically tractable with the introduction of strong simplifications, ignoring the involved biophysicalfeatures of the single units. Others are very faithful to reality at the price of very complex mathematical descriptions. Models are then used to
describe networks, often through simulations. The appearance of particular patterns in the trains of spikes is then used to compare with observed data or to switch from microscopic to macroscopic analysis. In this framework, it is essential to guarantee the reliability of the output of the model and often scientists compare the output of the models with real data. However, when the models are used to reproduce networks the output of some neurons becomes the input of a successive layer of neurons. This fact opens a problem of consistency between input and output of layers of neurons. To the best of our knowledge, this problem has not yet been deeply investigated. Here, we consider the simplest Stochastic Integrate and Fire model and we try to characterize the features of its input in such a way to re-obtain the same features in the output. In particular, we focus on the tail properties of ISIs distribution. Observed data suggest the presence of heavy tails for this distribution. Using the Stochastic Integrate and Fire paradigm for the neurons of the network we study how such features can be transmitted from the network. In this framework we introduce a particular class of multivariate distributions, i.e. the regularly varying distributions. We show that assuming that the input to a neuron is a regularly varying random vector and that successive ISIs of a neuron are asymptotically independent thenthe resulting output is a regularly varying random variable. The talk is based on a joint work with Petr Lansky and Federico Polito.

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We introduce the theory of semi-Markov processes and their interplay with non-local equations. Particular attention will be devoted to the theory of exit times from sets and the connection with boundary value problems, e.g., on a time-dependent parabolic domain.

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A fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. A similar definition is proposed for the (nonhomogeneous) fractional compound Poisson process. Both finite-dimensional and functional limit theorems are presented for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Some of the limiting results are verified in a Monte Carlo simulation.

Papers:

[1] Nikolai Leonenko, Enrico Scalas and Mailan Trinh, The fractional non-homogeneous Poisson process. Statistics and Probability Letters, 120, 2017, pp. 147-156. DOI: http://dx.doi.org/10.1016/j.spl.2016.09.024

https://arxiv.org/abs/1601.03965

[2] Nikolai Leonenko, Enrico Scalas and Mailan Trinh, Limit theorems for the fractional nonhomogeneous Poisson process, Journal of Applied Probability , 56:1, 2019 , pp. 246 - 264. DOI: https://doi.org/10.1017/jpr.2019.16

https://arxiv.org/abs/1711.08768

This is joint work with Nikolai Leonenko and Mailan Trinh.

##### Abstract:

In this talk we focus on a class of semi-Markov birth-death processes obtained by means of a time-change of some standard birth-death process. Precisely, we consider as parent processes the immigration-death process and the Meixner process, whose stationary distributions are respectively the Poisson and the Pascal distributions. Exploiting, on one hand, the properties of the Charlier and Meixner polynomials (in particular, the self-duality property), while, on the other, characterizing the eigenfunctions of some non-local operators by means of the Laplace transform of an inverse subordinator, we are able to explicitly express the spectral decomposition of the transition probability function of the aforementioned processes. The latter expression is then used to prove existence and uniqueness of strong solutions for a class of time-nonlocal Cauchy problems in a suitable Banach sequence space and the probabilistic interpretation of such equations as some sort of non-local backward/forward Kolmogorov equations. Finally, a comparison with the time-changed diffusion case is carried out by referring to the spectral decomposition of the probability density function of time-changed Pearson diffusions. The latter argument hints at the possibility of applying this kind of spectral methods to a wider range of problems.

This is the result of joint work with Nikolai Leonenko from Cardiff University and Enrica Pirozzi from University of Naples.

##### Abstract:

Exploration via random walks is often very useful for the analysis of directed networks. For instance, the walk's stationary distribution plays a prominent role in ranking systems and search algorithms. In this lecture, we present some recent progress in the analysis of random walks for a class of sparse directed networks generated by the so-called configuration model. We discuss various properties of the stationary distribution, including bulk behaviour and extremal values. We also consider the mixing time, that is the time needed to reach stationarity, and show that the walk typically displays a cutoff behaviour. Moreover, we discuss the asymptotic behaviour of the cover time, that is the expected time it takes the random walk to cover the whole network. Finally, we analyse the convergence to stationarity when the walk experiences regeneration events such as teleportation, as in the PageRank algorithm, or resampling of the underlying graph, as in a dynamically evolving network.

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Internal DLA is a mathematical model for the growth of a random cluster of particles according to the harmonic measure on the cluster boundary, seen from an internal point. Performing IDLA on cylinder graphs (i.e. graphs of the form $G \times \mathbb{Z}$) gives a Markov chain on the infinite space of particle configurations. We show that this chain is positive recurrent, and give a description of typical (i.e. stationary) configurations. We then analyse the mixing time of this chain.

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The first part of the talk is devoted to a brief introduction to supervised learning focusing on the regularised empirical risk minimization (ERM) on Reproducing Kernel Hilbert spaces. Though ERM achieves optimal convergence rates [1], it requires huge computational resources on high dimensional datasets. The second half of the talk is devoted to discuss some recent ideas where the hypothesis space is a low dimensional random space. This approach naturally leads to computational savings, but the question is whether the corresponding learning accuracy is degraded. If the random subspace is spanned by a random subset of the data, the statistical-computational tradeoff has been first explored for the least squares loss [2,3], for the least squares loss, then for self-concordant loss functions [4] , as the logistic loss, and, quite recently, for non-smooth convex Lipschitz loss functions [5], as the hinge loss.

References:

[1] Caponnetto, A. and De Vito, E. (2007). Optimal rates for the regularized least-squares algorithm. Foundations of Computational Mathematics, 7(3):331–368.

[2] Rudi, A., Calandriello, D., Carratino, L., and Rosasco, L. (2018). On fast leverage score sampling and optimal learning. In Advances in Neural Information Processing Systems, pages 5672–5682.

[3] Rudi, A., Camoriano, R., and Rosasco, L. (2015). Less is more: Nystr̈om computational regularization. In Advances in Neural Information Processing Systems, pages 1657–1665.

[4] Marteau-Ferey, U., Ostrovskii, D., Bach, F., and Rudi, A. (2019). Beyond least-squares: Fast rates for regularized empirical risk minimization through self-concordance. arXiv preprint arXiv:1902.03046.

[5] Andrea Della Vecchia, Jaouad Mourtada, Ernesto De Vito, Lorenzo Rosasco, Regularized ERM on random subspaces arXiv:2006.10016

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In this talk we present a rather flexible and expressive model for non-negative functions. We will show direct applications in probability representation and non-convex optimization. In particular, the model allows to derive an algorithm for non-convex optimization that is adaptive to the degree of differentiability of the objective function and achieves optimal rates of convergence. Finally, we show how to apply the same technique to other interesting problems in applied mathematics that can be easily expressed in terms of inequalities.

References:

Ulysse Marteau-Ferey , Francis Bach, Alessandro Rudi. Non-parametric Models for Non-negative Functions. https://arxiv.org/abs/2007.03926

Alessandro Rudi, Ulysse Marteau-Ferey, Francis Bach. Finding Global Minima via Kernel Approximations. https://arxiv.org/abs/2012.11978

##### Abstract:

In this talk I present a survey of results about linear and nonlinear equations of Kolmogorov type arising in physics and in mathematical finance. These equations typically satisfy a parabolic Hormander condition and induce rich intrinsic geometric structures. Results about the existence and regularity of solutions are presented. Recent applications to stochastic filtering are also discussed.

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In this talk we consider a class of SDEs with drift depending on the law density of the solution, known as McKean SDEs. The novelty here is that the drift is singular in the sense that it is `multiplied' by a generalised function (element of a negative fractional Sobolev space). Those equations are interpreted in the sense of a suitable singular martingale problem, thus a key tool is the study of the corresponding singular Fokker-Planck equation. We define the notion of solution to the singular McKean equation and show its existence and uniqueness. This is based on a joint work with F. Russo (ENSTA).

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The study of time change lays at the intersection of probability and statistics and have interesting potential in the modelling of different phenomena. These models are appealing, since they seem quite close to classical Lévy structures, often easy to simulate, though they are still statistically very different, as they may loose important properties, such as independent increments and Markovianity. It clearly all depends on the time change applied! When it comes to stochastic calculus, stochastic dynamics, and control, we shall see how the use of the interplay of partial information techniques and enlargement of filtrations can help dealing with such dynamics.

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We study the optimal proportional reinsurance and investment strategy for an insurance company which experiences both ordinary and catastrophic claims and wishes to maximize the expected exponential utility of its terminal wealth. We propose a model where the insurance framework is affected by environmental factors, and aggregate claims and stock prices are subject to common shocks, i.e. drastic events such as earthquakes, extreme weather conditions, or even pandemics, that have an immediate impact on the financial market and simultaneously induce insurance claims. Using the classical stochastic control approach based on the Hamilton-Jacobi- Bellman equation, we provide a verification result for the value function via classical solutions to two backward partial differential equations and characterize the optimal strategy. Finally, we discuss the effect of the common shock dependence via a comparison analysis.

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Consider a random variable $F$ associated with a random point configuration generated by a Poisson point process (for instance, $F$ is the length of some random geometric graph). How can we assess the discrepancy between the distribution of $F$ and the one of a Gaussian random variable? The aim of my talk is to describe some recent results in this direction, all relying on some quantitative version of the notion of "geometric stabilization", as formalized by Penrose & Yukich in 2001. In doing so, I will sketch the general philosophy of a powerful collection of techniques for establishing probabilistic approximations, known as the "Malliavin-Stein method". Some explicit examples will be discussed, in particular, related to models of combinatorial optimization and random forests.

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We will recall generalities on completely random measures -alias independently scattered measures- and their natural interaction with divergence-less flows on the underlying domain. We will then proceed to argue how these random fields thus constitute invariant measures for 2d incompressible Euler's equations, outline how a notion of solution to the latter nonlinear dynamics can be given in this setting by means of stochastic integrals, and discuss both rigorous results and open problems relating to the well-posedness of this measure-preserving system.

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I will present some recent convergence results toward Gaussian processes, that arise from statistical mechanics models and stochastic PDEs connected to the so-called KPZ equation. These results may be viewed as generalised central limit theorems and they can be proved with a blend of old and new techniques, whose wide interest I will try to illustrate.

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In this talk we investigate the behavior of the "typical" eigenfunction of a compact Riemannian manifold. In particular, motivated by both Yau's conjecture on nodal sets and Berry's ansatz on planar random waves, we consider random spherical harmonics and study the distribution of the length of their nodal lines for large eigenvalues. These results raise several questions regarding both the distribution of other geometric functionals of excursion sets at any level and the behavior of nodal statistics of random eigenfunctions of a "generic" manifold. In this talk we answer some of these questions, relying on recent developments in the theory of local geometry of random fields and Gaussian Kinematic Formulae à la Adler & Taylor.

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In this talk we present a market model including financial assets and life insurance liabilities within a reduced-form framework under model uncertainty by following [1]. In particular we extend this framework to include mortality intensities following an affine process under parameter uncertainty, as defined in [2]. This allows both to introduce the definition of a longevity bond under model uncertainty in a consistent way with the classical case under one prior, as well as to compute it by explicit formulas or by numerical methods. We also study conditions to guarantee the existence of a càdlàg modification for the longevity bond’s value process. Furthermore, we show how the resulting market model extended with the longevity bond is arbitrage-free and study arbitrage-free pricing of contingent claims or life insurance liabilities in this setting. This talk is based on: [1] Francesca Biagini and Yinglin Zhang. Reduced-form framework under model uncertainty. The Annals of Applied Probability, 29(4):2481–2522, 2019. [2] Francesca Biagini and Katharina Oberpriller. Reduced-form framework under model uncertainty. Preprint University of Munich and Gran Sasso Science Institute, 2020. [3] Tolulope Fadina, Ariel Neufeld, and Thorsten Schmidt. Affine processes under parameter uncertainty. Probability, Uncertainty and Quantitative Risk volume 4 (5), 2019.

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We give conditions ensuring that the backward partial integro differential equation (PIDE) associated with a multidimensional jump-diffusion with a pure jump component has a unique classical solution. Our proof uses a probabilistic arguments and extends the results of (Pham 1998) to processes with a pure jump component where the jump intensity is modulated by a diffusion process. This result is particularly useful in some applications to pricing and hedging of financial and actuarial instruments, and we provide an example to pricing of catastrophic (CAT) bonds.

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In the first part of this talk I will present a survery on the relationships among classical stochastic optimal control problems, non-linear partial differential equations (the Hamilton-Jacobi-Bellman equations) and backward stochastic differential equations (BSDEs). In the second part, more specifically, I will introduce the so-called randomization method, which allows to associate an appropriate BSDE to a large class of optimal control problems. Among the possible generalizations, I will concentrate on optimal control of path-dependent equations, i.e. equations with general with memory effects.

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In the present talk I will study a stochastic optimal control problem with path-dependent coefficients. I will exploit the so-called randomization method to derive a dynamic programming principle for the value function. This allows to prove that the value function is a viscosity solution to a path-dependent Hamilton-Jacobi-Bellman equation, involving the horizontal and vertical derivatives of functional Ito calculus. Finally, I will discuss the validity of the comparison principle for such a partial differential equation.

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Random resistor networks allow to study electron transport in disordered systems. By means of stochastic homogenization one can characterize the infinite volume effective conductivity in terms of the homogenized matrix of suitable discrete Poisson equations. We will then focus on the low temperature behavior of the random conductance model and the Miller-Abrahams random resistor network. The latter models the so-called Mott’s variable range hopping, a fundamental transport mechanism in disordered solids in the regime of strong Anderson localization. We will discuss its percolative properties and show how homogenization, percolation and scaling arguments lead to Mott’s law. (For the percolation part, the talk is based on joint works with H.A. Mimun).

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Link.

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We consider two first-passage percolation processes, $FPP_1$ and $FPP_\lambda$, spreading with rates $1$ and $\lambda$ respectively, on a graph $G$ with bounded degree. $FPP_1$ starts from a single source, while the initial configuration of $FPP_\lambda$ consists of countably many seeds distributed according to a product of iid Bernoulli random variables of parameter $\mu$ on the set of vertices. This model is known as "First passage percolation in a hostile environment" (FPPHE), and was introduced by Stauffer and Sidoravicius as an auxiliary model for investigating a notoriously challenging model called Multiparticle Diffusion Limited Aggregation. We consider several questions about FPPHE, focusing on the case when $G$ is a non-amenable hyperbolic graph. This talk is based on joint works with Alexandre Stauffer.

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The presence of the irregular fluctuations of a noise sometimes improves the theory of differential equations, ordinary or partial, a phenomenon today called "regularization by noise". This joint talk will introduce the problem in finite dimensions, by some classical and some more recent examples and results. Then it moves to SPDEs, where the results are mostly recent and under investigation. The different roles of additive noise and multiplicative transport type noise, the latter with its fluid mechanic flavour, will be described.