Since the seminal work of Wiener (Am J Math 60:897–936, 1938), chaos expansion has evolved to a powerful methodology for studying a broad range of stochastic differential equations. Yet its complexity for systems subject to the white noise remains significant. The issue appears due to the fact that the random increments generated by the Brownian motion result in a growing set of random variables with respect to which the process could be measured. In order to cope with this high dimensionality, we present a novel transformation of stochastic processes driven by the white noise. In particular, we show that under suitable assumptions, the diffusion arising from white noise can be cast into a logarithmic gradient induced by the measure of the process. Through this transformation, the resulting equation describes a stochastic process whose randomness depends only on the initial condition. Therefore, the stochasticity of the transformed system lives in the initial condition and it can be treated conveniently with chaos expansion tools.

In this paper, we study a robust estimation method for the observation-driven integer-valued time-series models in which the conditional probability mass of current observations is assumed to follow a negative binomial distribution. Maximum likelihood estimator is highly affected by the outliers. We resort to the minimum density power divergence estimator as a robust estimator and show that it is strongly consistent and asymptotically normal under some regularity conditions. Simulation results are provided to illustrate the performance of the estimator. An application is performed on data for campylobacteriosis infections.

For the multiplicative background risk model, a distortion-type risk measure is used to measure the tail risk of the portfolio under a scenario probability measure with multivariate regular variation. In this paper, we investigate the tail asymptotics of the portfolio loss $\sum _{i=1}^{d}R_iS$, where the stand-alone risk vector ${\mathbf {R}}=(R_1,\ldots ,R_d)$ follows a multivariate regular variation and is independent of the background risk factor S. An explicit asymptotic formula is established for the tail distortion risk measure, and an example is given to illustrate our obtained results.

In this paper, we prove a Chern number inequality for Higgs bundles over some Kähler manifolds. As an application, we get the Bogomolov inequality for semi-stable parabolic Higgs bundles over smooth projective varieties.

We show that the normalized cochain complex of a nonsymmetric cyclic operad with multiplication is a Quesney homotopy BV algebra; as a consequence, the cohomology groups form a Batalin–Vilkovisky algebra, which is a result due to L. Menichi. We provide ample examples.

Finding optimal knots is a challenging problem in spline fitting due to a lack of prior knowledge regarding optimal knots. The unimodality of initial B-spline approximations associated with given data is a promising characteristic of locating optimal knots and has been applied successfully. The initial B-spline approximations herein are required to approximate given data well enough and characterized by the unimodality if jumps from the highest-order derivatives of the approximations at some interior knots are local maxima. In this paper, we prove the unimodality of the initial B-spline approximations that are constructed under two assumptions: Data points are sampled uniformly and sufficiently from B-spline functions, and initial knots are chosen as the parameters of sampling points. Our work establishes the theoretical basis of the unimodality of initial B-spline approximations and pioneers the theoretical study of locating optimal knots.

The algebraic structure of skew left brace has proved to be useful as a source of set-theoretic solutions of the Yang–Baxter equation. We study in this paper the connections between left and right $\pi $-nilpotency and the structure of finite skew left braces. We also study factorisations of skew left braces and their impact on the skew left brace structure. As a consequence of our study, we define a Fitting-like ideal of a left brace. Our approach depends strongly on a description of a skew left brace in terms of a triply factorised group obtained from the action of the multiplicative group of the skew left brace on its additive group.