LiBERA On-line Seminars

Relying on the backward Kolmogorov PDE stated on the Wasserstein space, we obtain several new results concerning the approximation error of some non-linear diffusion process in the sense of McKean-Vlasov by the corresponding Euler-Maruyama discretization scheme of its system of interacting particles. We notably present explicit error estimates, at the level of the trajectories, at the level of the semigroup (stated on the Wasserstein space) and at the level of the densities. Some Gaussian density estimates of the transition density and its first order derivative for the Euler-Maruyama scheme are also established. This presentation is based on joint works with Clément Rey (Ecole Polytechnique) and Xuanye Song (Université Paris Cité).

In this talk we start with an elementary, but useful quantitative generalization of the first Borel-Cantelli lemma. The idea is to translate good rate of convergence of probability into higher order moments of the a.s. modulus of convergence. Our results quantify almost sure convergence in terms of the total number of occurrences of the error events, and the last index if its occurrence. The talk is motivated by different martingale convergence results such as the convergence of Polya urns, L^p or exponentially bounded martingales etc. We provide a sample of applications, such as the strong law of large numbers, the presence of a large deviations principle, the method of moments in statistics and numerical analysis.

This is joint work with A. Steinicke (MU Leoben, Austria) and Luisa F. Estrada (U. Warwick, UK).

The aim of this work is to determine the optimal cyber-security investment strategy for an entity subject to cyber-attacks. Inspired by the Gordon-Loeb model, we assume that the success rate of cyber-attacks depends on the vulnerability of the security system under threat, which can be reduced investing in IT security measures. We introduce a dynamic version of the Gordon-Loeb setting, by exploiting Hawkes processes to model the attacks' arrival.  This dynamic framework is crucial to rapidly react to the  changes which characterize cyber-risk. The problem is framed as a Markovian 2-dimensional stochastic control problem with jumps and it is addressed using dynamic programming techniques. The optimal value is characterized by a partial integro-differential equation, solved numerically. The corresponding optimal strategy is explicitly obtained by differentiating the optimal value function. Work in progress with C. Fontana, C. Hillairet and B. Ongarato.

In this presentation we study, as far as possible, the convergence to equilibrium as the noise intensity $\epsilon$ tends to zero for ergodic random systems out of equilibrium. Particularly, we focus in the so-called Cox-Ingersoll-Ross (CIR) process, which is governed by a stochastic differential equation with multiplicative non-linear noise of the form:
\[
d X^{\epsilon}_t(x)=(b-aX^{\epsilon}_t(x))dt+\epsilon \sqrt{X^{\epsilon}_t(x)}dB_t,\quad X_0=x,
\]
where $x>0$, $a>0$ and $b>0$ are constants, and $(B_t)_{t>0}$ is a one-dimensional standard Brownian motion.
 
Specifically, we establish the occurrence of the so-called profile cutoff phenomenon in the total variation distance and in the renormalized Wasserstein distance when the intensity of the noise tends to zero. Our results include explicit cut-off time, explicit time window, and explicit profile function. Furthermore, asymptotics of the so-called mixing times are given explicitly. This is based in the paper Profile cut-off phenomenon for the ergodic Feller root process, SPA 2025, with Liliana Esquivel, Universidad de Puerto Rico, Puerto Rico.
 
If time permits, I will discuss further extensions of this framework to continuous-state branching processes with immigration (CBI), which is an ongoing work with Juan Carlos Pardo, CIMAT, Mexico.