CyberSec Research

We prove comparison theorems for the horizontal Laplacian of the Riemannian distance in the context of Riemannian foliations with minimal leaves. This general framework generalizes previous works and allow us to consider the sub-Laplacian of Carnot groups of arbitrary steps. The comparison theorems yield a Bonnet-Myers type theorem, stochastic completeness and Lipschitz regularization property for the sub-Riemannian semigroup.

Spectral properties of Toeplitz operators and their finite truncations have long been central in operator theory. In the finite dimensional, non-normal setting, the spectrum is notoriously unstable under perturbations. Random perturbations provide a natural framework for studying this instability and identifying spectral features that emerge in typical noisy situations. This article surveys recent advances on the spectral behavior of (polynomially vanishing) random perturbations of Toeplitz matrices, focusing mostly on the limiting spectral distribution, the distribution of outliers, and localization of eigenvectors, and highlight the major techniques used to study these problems. We complement the survey with new results on the limiting spectral distribution of polynomially vanishing random perturbation of Toeplitz matrices with continuous symbols, on the limiting spectral distribution of finitely banded Toeplitz matrices under exponentially and super-exponentially vanishing random perturbations, and on the complete localization of outlier eigenvectors of randomly perturbed Jordan blocks.

Fix a smooth Morse function $U\colon \mathbb{R}^{d}\to\mathbb{R}$ with finitely many critical points, and consider the solution of the stochastic differential equation \begin{equation*} d\bm{x}_{\epsilon}(t)=-\nabla U(\bm{x}_{\epsilon}(t))\,dt \,+\,\sqrt{2\epsilon}\, d\bm{w}_{t}\,, \end{equation*} where $(\bm{w}_{t})_{t\ge0}$ represents a $d$-dimensional Brownian motion, and $\epsilon>0$ a small parameter. Denote by $\mathcal{P}(\mathbb{R}^{d})$ the space of probability measures on $\bb R^d$, and by $\mathcal{I}_{\epsilon} \colon \mathcal{P}(\mathbb{R}^{d})\to[0,\,\infty]$ the Donsker--Varadhan level two large deviations rate functional. We express $\mc I_\epsilon$ as $\mc I_\epsilon = \epsilon^{-1} \mc J^{(-1)} + \mc J^{(0)} + \sum_{1\le p\le \mf q} (1/\theta^{(p)}_\epsilon) \, \mc J^{(p)}$, where $\mc J^{(p)}\colon \mc P(\bb R^d) \to [0,+\infty]$ stand for rate functionals independent of $\epsilon$ and $\theta^{(p)}_\epsilon$ for sequences such that $\theta^{(1)}_\epsilon \to\infty$, $\theta^{(p)}_\epsilon / \theta^{(p+1)}_\epsilon \to 0$ for $1\le p< \mf q$. The speeds $\theta^{(p)}_\epsilon$ correspond to the time-scales at which the diffusion $\bm{x}_{\epsilon}(\cdot)$ exhibits a metastable behaviour, while the functional $\mc J^{(p)}$ represent the level two, large deviations rate functionals of the finite-state, continuous-time Markov chains which describe the evolution of the diffusion $\bm{x}_{\epsilon}(\cdot)$ among the wells in the time-scale $\theta^{(p)}_\epsilon$.

In a Latin square, every row can be interpreted as a permutation, and therefore has a parity (even or odd). We prove that in a uniformly random $n\times n$ Latin square, the $n$ row parities are very well approximated by a sequence of $n$ independent unbiased coin flips: for example, the total variation error of this approximation tends to zero as $n\to\infty$. This resolves a conjecture of Cameron. In fact, we prove a generalisation of Cameron's conjecture for the joint distribution of the row parities, column parities and symbol parities (the latter are defined by the symmetry between rows, columns and symbols of a Latin square). Along the way, we introduce several general techniques for the study of random Latin squares, including a new re-randomisation technique via `stable intercalate switchings', and a new approximation theorem comparing random Latin squares with a certain independent model.

Rooted trees are essential for describing numerical schemes via the so-called B-series. They have also been used extensively in rough analysis for expanding solutions of singular Stochastic Partial Differential Equations (SPDEs). When one considers scalar-valued equations, the most efficient combinatorial set is multi-indices. In this paper, we investigate the existence of intermediate combinatorial sets that will lie between multi-indices and rooted trees. We provide a negative result stating that there is no combinatorial set encoding elementary differentials in dimension $d\neq 1$, and compatible with the rooted trees and the multi-indices aside from the rooted trees. This does not close the debate of the existence of such combinatorial sets, but it shows that it cannot be obtained via a naive and natural approach.

We consider local singular perturbations of a one-dimensional Laplace operator from the point of view of semigroup theory. Under certain assumptions, we prove the convergence of the corresponding semigroups to the heat semigroup with Robin-type boundary conditions at zero. As an application we provide semigroup theoretical approach to calculating the distribution of the local time for Brownian motion when it exits an interval.

Strassen's theorem asserts that for given marginal probabilities $\mu,\nu$ there exists a martingale starting in $\mu$ and terminating in $\nu$ if and only if $\mu,\nu$ are in convex order. From a financial perspective, it guarantees the existence of market-consistent martingale pricing measures for arbitrage-free prices of European call options and thus plays a fundamental role in robust finance. Arbitrage-free prices of American options demand a stronger version of martingales which are 'biased' in a specific sense. In this paper, we derive an extension of Strassen's theorem that links them to an appropriate strengthening of the convex order. Moreover, we provide a characterization of this order through integrals with respect to compensated Poisson processes.

This paper establishes the quantitative stability of invariant measures $\mu_{\alpha}$ for $\mathbb{R}^d$-valued ergodic stochastic differential equations driven by rotationally invariant multiplicative $\alpha$-stable processes with $\alpha\in(1,2]$. Under structural assumptions on the coefficients with a fixed parameter vector $\bm{\theta}$, we derive optimal convergence rates in the Wasserstein-$1$ ($\cW_{1}$) distance between the invariant measures introduced above, namely, \item[(i)] For any interval $[\alpha_0, \vartheta_0] \subset (1,2)$, there exists $C_1 = C(\alpha_0, \vartheta_0,\bm{\theta},d) > 0$ such that \cW_{1}(\mu_\alpha, \mu_\vartheta) \leq C_1 |\alpha - \vartheta|, \quad \forall \alpha, \vartheta \in [\alpha_0, \vartheta_0]. \item[(ii)] For any $\alpha_0\in (1,2)$, there exists $C_2 = C(\alpha_0, \bm{\theta}) > 0$ such that \begin{align*} \cW_{1}(\mu_\alpha, \mu_2) \leq C_2\, d(2 - \alpha), \quad \forall \alpha \in [\alpha_0, 2). The optimality of these rates is rigorously verified by explicit calculations for the Ornstein-Uhlenbeck systems in \cite{Deng2023Optimal}. It is worth emphasizing that \cite{Deng2023Optimal} addressed only case (ii) under additive noise, whereas our analysis establishes results for both cases (i) and (ii) under multiplicative $\alpha$-stable noise, employing fundamentally different analytical methods.

Classical filtrations in probability theory formalize the accumulation of information along a linear time axis: the past is unique and the present evolves into an uncertain future. In many contexts, however, information is neither linear nor uniquely determined by a single history. In this paper we propose a geometric model of synthetic filtrations, where the present may be formed by synthesizing multiple possible pasts. To achieve this, we introduce a new category {\Sigma}, an extension of the simplex category {\Delta}, whose objects encode context-dependent times. Synthetic filtrations are realized as contravariant functors {\Sigma} {\to} Prob, where Prob is the category of probability spaces with null-preserving maps. After seeing a general definition of synthetic filtrations, we define, as a concrete example, Dirichlet filtrations, where probability measures on simplices arise from Dirichlet distributions, highlighting the role of parameter and contextual uncertainty. Finally, we interpret Bayesian updating as a categorical update of a Dirichlet functor, showing how learning fits naturally into the synthetic filtration framework. This work combines categorical probability, simplicial geometry, and Bayesian statistics, and suggests new applications in finance, stochastic modeling, and subjective probability.

In this article, we propose a least squares method for the estimation of the transition density in bifurcating Markov models. Unlike the kernel estimation, this method do not use the quotient which can be a source of errors. In order to study the rate of convergence for least squares estimators, we develop exponential inequalities for empirical process of bifurcating Markov chain under bracketing assumption. Unlike the classical processes, we observe that for bifurcating Markov chains, the complexity parameter depends on the ergodicity rate and as consequence, we have that the convergence rate of our estimator is a function of the ergodicity rate. We conclude with a numerical study to validate our theoretical results.

Myopic optimization (MO) outperforms reinforcement learning (RL) in portfolio management: RL yields lower or negative returns, higher variance, larger costs, heavier CVaR, lower profitability, and greater model risk. We model execution/liquidation frictions with mark-to-market accounting. Using Malliavin calculus (Clark-Ocone/BEL), we derive policy gradients and risk shadow price, unifying HJB and KKT. This gives dual gap and convergence results: geometric MO vs. RL floors. We quantify phantom profit in RL via Malliavin policy-gradient contamination analysis and define a control-affects-dynamics (CAD) premium of RL indicating plausibly positive.

Consider the following stochastic reaction-diffusion equation with logarithmic superlinear coefficient b, driven by space-time white noise W: $$ u_t(t,x) = (1/2)u_{xx}(t,x) + b(u(t,x)) + \sigma(u(t,x))W(dt,dx) $$ for $t > 0$ and $x \in [0,1]$, with initial condition $$ u(0,x) = u_0(x) $$ for $x \in [0,1]$, where $u_0 \in L^2[0,1]$. In this paper, we establish existence and uniqueness of probabilistically strong solutions in $C(R_+, L^2[0,1])$. Our result resolves a problem from [Ann. Probab. 47 (2019) 519-559] and provides an alternative proof of the non-blowup of $L^2[0,1]$ solutions from the same reference. We use new Gronwall-type inequalities. Due to nonlinearity, we work with first order moments, requiring precise estimates of the stochastic convolution.

We introduce a mltiparameter version of Skellam point process via multiparameter Poisson processes. Its distributional properties are studied in detail. Its compound representation is derived for a particular case. Also, its Riemann integral over a rectangle in $\mathbb{R}^M_+$, $M\ge1$ is introduced and a closed expression for its characteristic function is obtained. Later, we introduce a different version of multiparameter Skellam process, and derive a weak convergence result for it. Moreover, a two parameter fractional Skellam process is discussed.

The Gillespie algorithm and its extensions are commonly used for the simulation of chemical reaction networks. A limitation of these algorithms is that they have to process and update the system after every reaction, requiring significant computation. Another class of algorithms, based on the tau-leaping method, is able to simulate multiple reactions at a time at the cost of decreased accuracy. We present a new algorithm for the exact simulation of chemical reaction networks that is capable of sampling multiple reactions at a time via a first-order approximation similarly to the tau-leaping methods. We prove that the algorithm has an improved runtime complexity compared to existing methods for the exact simulation of chemical reaction networks, and present an efficient and easy to use implementation that outperforms existing methods in practice.

In this paper, we provide a new property of value at risk (VaR), which is a standard risk measure that is widely used in quantitative financial risk management. We show that the subadditivity of VaR for given loss random variables holds for any confidence level if and only if those are comonotonic. This result also gives a new equivalent condition for the comonotonicity of random vectors.

This paper explores the rates of convergence of solutions for multivariate stochastic differential equations (SDEs) driven by L\'evy processes within the small-time stable domain of attraction (DoA). Explicit bounds are derived for the uniform Wasserstein distance between solutions of two L\'evy-driven SDEs, expressed in terms of driver characteristics. These bounds establish convergence rates in probability for drivers in the DoA, and yield uniform Wasserstein distance convergence for SDEs with additive noise. The methodology uses two couplings for L\'evy driver jump components, leading to sharp convergence rates tied to the processes' intrinsic properties.

Let $P_n(x) = \sum_{k=0}^{n} \xi_k x^k$ be a Kac random polynomial, where the coefficients $\xi_k$ are i.i.d.\ copies of a given random variable $\xi$. Based on numerical experiments, it has been conjectured that if $\xi$ has mean zero, unit variance, and a finite $(2+\varepsilon_0)$-moment for some $\varepsilon_0>0$, then \[ \mathbb{E}[N_{\mathbb{R}}(P_n)] \;=\; \frac{2}{\pi} \log n + C_{\xi} + o_n(1), \] where $N_{\mathbb{R}}(P_n)$ denotes the number of real roots of $P_n$, and $C_{\xi}$ is an absolute constant depending only on $\xi$, which is nonuniversal. Prior to this work, the existence of $C_{\xi}$ had only been established by Do-Nguyen-Vu (2015, \emph{Proc.\ Lond.\ Math.\ Soc.}) under the additional assumption that $\xi$ either admits a $(1+p)$-integrable density or is uniformly distributed on $\{\pm 1, \pm 2, \dots, \pm N\}$. In this paper, using a different method, we remove these extra conditions on $\xi$, and extend the result to the setting where the $\xi_k$ are independent but not necessarily identically distributed. Moreover, this proof strategy provides an alternative description of the constant $C_{\xi}$, and this new perspective serves as the key ingredient in establishing that $C_{\xi}$ depends continuously on the distribution of $\xi$.

We model the scrambling of a Rubik's cube by a Markov chain and introduce a stopping time $T$ which is a quite natural candidate to be a strong uniform time. This may pave the way for estimating the number of moves required to scramble a cube. Unfortunately, we show that $T$ is not this is not strongly uniform.