Let X be a submartingale starting from 0, and Y be a semimartingale which is orthogonal and strongly differentially subordinate to X. The paper contains the proof of the sharp estimate $\mathbb{P}\left(su{p}_{t\ge 0}\right|Y_t\left|\ge 1\right)\le 3.375...\parallel X\parallel ₁$. As an application, a related weak-type inequality for smooth functions on Euclidean domains is established.

Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $\parallel Y{\parallel }_W\le \left(2(\alpha +1)²\right)/(2\alpha +1)\parallel X{\parallel }_{L^{\infty }}$. Here W is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667–672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis...

Weighted Gamma (WG), a weighted version of Gamma distribution, is introduced. The hazard function is increasing or upside-down bathtub depending upon the values of the parameters. This distribution can be obtained as a hidden upper truncation model. The expressions for the moment generating function and the moments are given. The non-linear equations for finding maximum likelihood estimators (MLEs) of parameters are provided and MLEs have been computed through simulations and also for a real data...

In this paper, we prove the existence and uniqueness of the solution of the initial boundary value problem for a stochastic mass conserved Allen-Cahn equation with nonlinear diffusion together with a homogeneous Neumann boundary condition in an open bounded domain of ${ℝ}^n$ with a smooth boundary. We suppose that the additive noise is induced by a Q-Brownian motion.

Let $\left(X_j\right)$ be a sequence of independent identically distributed random operators on a Banach space. We obtain necessary and sufficient conditions for the Abel means of $X_n...X_2{X}_1$ to belong to Hardy and Lipschitz spaces a.s. We also obtain necessary and sufficient conditions on the Fourier coefficients of random Taylor series with bounded martingale coefficients to belong to Lipschitz and Bergman spaces.

We offer the quantitative estimation of stability of risk-sensitive cost optimization in the problem of optimal stopping of Markov chain on a Borel space $X$. It is supposed that the transition probability $p(·|x)$, $x\in X$ is approximated by the transition probability $\tilde{p}(·|x)$, $x\in X$, and that the stopping rule ${\tilde{f}}_{*}$ , which is optimal for the process with the transition probability $\tilde{p}$ is applied to the process with the transition probability $p$. We give an upper bound (expressed in term of the total variation distance: ${sup}_{x\in X}\parallel p(·|x)-\tilde{p}(·|x)\parallel )$ for...

In this paper, we consider a symmetric α-stable p-sub-stable two-dimensional random vector. Our purpose is to show when the function $exp-\left(\right|a|p+{\left|b\right|p)}^{\alpha /p}$ is a characteristic function of such a vector for some p and α. The solution of this problem we can find in [3], in the language of isometric embeddings of Banach spaces. Our proof is based on simple properties of stable distributions and some characterization given in [4].