Related papers: Lecture notes on martingale inequalities
In a context of illiquidity, the reservation price is a well-accepted alternative to the usual martingale approach which does not apply. However, this price is not available in closed form and requires numerical methods such as Monte Carlo…
Some mathematical inequalities among various weighted means are studied. Inequalities on weighted logarithmic mean are given. Besides, the gap in Jensen's inequality is studied as a convex function approach. Consequently, some non-trivial…
Linear programming (LP) is an extremely useful tool and has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such…
Researchers have widely used exploratory factor analysis (EFA) to learn the latent structure underlying multivariate data. Rotation and regularised estimation are two classes of methods in EFA that they often use to find interpretable…
Some q-analysis variants of Hardy type inequalities of the form \int_0^b (x^{\alpha-1} \int_0^x t^{-\alpha} f(t) d_qt)^p d_qx \leq C \int_0^b f^p(t) d_qt with sharp constant C are proved and discussed. A similar result with the…
We prove quenched~$L^p$--type estimates for the gradient of a solution of a quasilinear elliptic equation with random coefficients.
We suggest two versions of the Hardy--Littlewood--Sobolev inequality for discrete time martingales. In one version, the fractional integration operator is a martingale transform, however, it may vanish if the filtration is excessively…
In the paper, we describe in operator form classes of PDEs that admit PINN's error estimation. Also, for $L^p$ spaces, we obtain a Bramble-Hilbert type lemma that is a tool for PINN's residuals bounding.
This survey provides an exposition of a suite of techniques based on the theory of polynomials, collectively referred to as polynomial methods, which have recently been applied to address several challenging problems in statistical…
We study mapping properties of finite field k-plane transforms. Using geometric combinatorics, we do an elaborate analysis to recover the critical endpoint estimate. As a consequence, we obtain optimal L^p-L^r estimates for all k-plane…
We establish $L^p$ Sobolev mapping properties for averages over certain curves in $\R^3$, which improve upon the estimates obtained by $L^2-L^\infty$ interpolation.
While multilevel Monte Carlo (MLMC) methods for the numerical approximation of partial differential equations with random coefficients enjoy great popularity, combinations with spatial adaptivity seem to be rare. We present an adaptive MLMC…
Using changes of probability measure developed by \mbox{Grama} and Haeusler (Stochastic Process.\ Appl., 2000), we obtain two generalizations of the deviation inequalities of Lanzinger and Stadtm\"{u}ller (Stochastic Process.\ Appl., 2000)…
Machine learning interatomic potentials (MLIPs) often neglect long-range interactions, such as electrostatic and dispersion forces. In this work, we introduce a straightforward and efficient method to account for long-range interactions by…
We propose a new weak convergence theorem for martingales, under gentler conditions than the usual convergence in probability of the sequence of associated quadratic variations. Its proof requires the combined use of Skorohod's…
A statistical mechanics argument relating partition functions to martingales is used to get a condition under which random geometric processes can describe interfaces in 2d statistical mechanics at criticality. Requiring multiple SLEs to…
Linear time-invariant systems are very popular models in system theory and applications. A fundamental problem in system identification that remains rather unaddressed in extant literature is to leverage commonalities amongst related linear…
These Lecture Notes are a brief introduction to the Malliavin calculus. In particular, different notions of Malliavin derivative found in the literature are considered and compared.
We give concentration bounds for martingales that are uniform over finite times and extend classical Hoeffding and Bernstein inequalities. We also demonstrate our concentration bounds to be optimal with a matching anti-concentration…
We provide a simple method and relevant theoretical analysis for efficiently estimating higher-order lp distances. While the analysis mainly focuses on l4, our methodology extends naturally to p = 6,8,10..., (i.e., when p is even).…