Related papers: Lecture notes on martingale inequalities
We prove oracle inequalities for a penalized log-likelihood criterion that hold even if the data are not independent and not stationary, based on a martingale approach. The assumptions are checked for various contexts: density estimation…
We consider functions L_p-integrable with Jacobi weights on [-1,1] and prove Hardy--Littlewood type inequalities for fractional integrals. As applications, we obtain the sharp (L_p, L_q) Ulyanov-type inequalities for the Ditzian--Totik…
In this paper, based on the techniques of Malliavin calculus, we provide some new concentration inequalities for the running supremum of the It\^o stochastic integral with unbounded integrands. Several applications and examples are provided…
This article introduces the splitting method to systems responding to rough paths as external stimuli. The focus is on nonlinear partial differential equations with rough noise but we also cover rough differential equations. Applications to…
We propose an inference procedure for estimators defined by mathematical programming problems, focusing on the important special cases of linear programming (LP) and quadratic programming (QP). In these settings, the coefficients in both…
We consider additive functionals of stationary Markov processes and show that under Kipnis-Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Levy area that can be…
These notes concern linear transformations on R^n and C^n, exponentials of linear transformations, and some related geometric questions.
We generalize a famous tail Doob's inequality, relative two non-negative random variables, arising in the martingale theory, in two directions: on the more general source data and on the random variables belonging to the so-called Grand…
This paper is devoted to the study of dispersive estimates for matrix Schr\"odinger equations on the half-line with general boundary condition, and on the line. We prove $L^{p}-L^{p^{\prime}}$ estimates on the half-line for slowly decaying…
We prove an elementary yet useful inequality bounding the maximal value of certain linear programs. This leads directly to a bound on the martingale difference for arbitrarily dependent random variables, providing a generalization of some…
An efficient computational algorithm to price financial derivatives is presented. It is based on a path integral formulation of the pricing problem. It is shown how the path integral approach can be worked out in order to obtain fast and…
We introduce the mixed-norm amalgam spaces $(L^{\vec{p}},L^{\vec{s}})(\mathbb{R}^n)$ and $(L^{\vec{p}},L^{\vec{s}})^{\alpha}(\mathbb{R}^n)$, and show their some basic properties. In addition, we find the predual…
Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems that related to integral estimates and regularity of solutions to the elliptic…
In this paper we come up with a dual version of the Furstenberg problem and obtain partial results via $L^p$ estimates of orthogonal projections. Examples are also discussed. Moreover, compared with general sets, we find that special…
We discuss the asymptotic behaviour for the best constant in L^p-L^q estimates for trigonometric polinomials and for an integral operator which is related to the solution of inhomogeneous Schrodinger equations. This gives us an opportunity…
This paper introduces Martingales by covering introductory measure theory concepts and the Lebesgue Integration and Conditional Expectation. It follows up with proofs of Kolomorgov's Theorem on conditional expectations, the Martingale…
We prove maximal inequalities for $L_q$-valued martingales obtained by stochastic integration with respect to compensated random measures. A version of these estimates for integrals with respect to compensated Poisson random measures were…
We extend some results about F\"ollmer's pathwise It\^o calculus that have only been derived for continuous paths to c\`adl\`ag paths with quadratic variation. We study some fundamental properties of pathwise It\^o integrals with respect to…
As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a stochastic maximal inequality derived by using the Taylor expansion, is…
For the homogeneous Boltzmann equation with (cutoff or non cutoff) hard potentials, we prove estimates of propagation of Lp norms with a weight $(1+ |x|^2)^q/2$ ($1 < p < +\infty$, $q \in \R\_+$ large enough), as well as appearance of such…