Related papers: On some estimates for bounded submartingales and t…
In this paper we study upper and lower bounds on the Bregman divergence $\Delta_{\mathcal{F}}^{\xi}(y,x):=\mathcal{F}(y)-\mathcal{F}(x)-\langle \xi, y-x\rangle $ for some convex functional $\mathcal{F}$ on a normed space $\mathcal{X}$, with…
We consider SDEs of the form $dX_t = |f(X_t)|/t^{\gamma} dt+1/t^{\gamma} dB_t$, where $f(x)$ behaves comparably to $|x|^k$ in a neighborhood of the origin, for $k\in [1,\infty)$. We show that there exists a threshold value…
It has been recently discovered that a convex function can be determined by its slopes and its infimum value, provided this latter is finite. The result was extended to nonconvex functions by replacing the infimum value by the set of all…
We give a slight improvement of the best known lower bound for the supremum of autoconvolutions of nonnegative functions supported in a compact interval. Also, by means of explicit examples we disprove a long standing natural conjecture of…
The softmax function is a ubiquitous component at the output of neural networks and increasingly in intermediate layers as well. This paper provides convex lower bounds and concave upper bounds on the softmax function, which are compatible…
In this paper, we study a delayed forward-backward stochastic control system in which all the coefficients depend on the state and control terms, and the control domain is not necessarily convex. A global stochastic maximum principle is…
In this paper upper bounds are given for the successive differences $A_{n+1}-A_{n}$ and B$_{n}-B_{n-1}$ where $A_{n}=1/(n-1) \tsum_{r=1}^{n-1}f(r/n)$, $B_{n}=1/(n+1) \tsum_{r=0}^{n}f(r/n)$ and $f$ is superquadratic function. We obtain…
The dominated convergence theorem implies that if (f_n) is a sequence of functions on a probability space taking values in the interval [0,1], and (f_n) converges pointwise a.e., then the sequence of integrals converges to the integral of…
We propose a method to bound the expectation of the supremum of the price process in stochastic volatility models. It can be applied, for example, to the rough Bergomi model, avoiding the need to discuss finiteness of higher moments. Our…
We present a set of high-probability inequalities that control the concentration of weighted averages of multiple (possibly uncountably many) simultaneously evolving and interdependent martingales. Our results extend the PAC-Bayesian…
Let $X$ and $Y$ denote two independent squared Bessel processes of dimension $m$ and $n-m$, respectively, with $n\geq 2$ and $m \in [0, n)$, making $X+Y$ a squared Bessel process of dimension $n$. For appropriately chosen function $s$, the…
We consider the Chance Constrained Model Predictive Control problem for polynomial systems subject to disturbances. In this problem, we aim at finding optimal control input for given disturbed dynamical system to minimize a given cost…
In this note we prove that the local martingale part of a convex function f of a d-dimensional semimartingale X = M + A can be written in terms of an It^o stochastic integral \int H(X)dM, where H(x) is some particular measurable choice of…
A general maximum principle (necessary and sufficient conditions) for an optimal control problem governed by a stochastic differential equation driven by an infinite dimensional martingale is established. The solution of this equation takes…
We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the…
Let $M_n= \fsu X1n$ be a sum of independent random variables such that $ X_k\leq 1$, $\E X_k =0$ and $\E X_k^2=\s_k^2$ for all $k$. Hoeffding 1963, Theorem 3, proved that $$\P{M_n \geq nt}\leq H^n(t,p),\quad H(t,p)= \bgl(1+qt/p\bgr)^{p +qt}…
Stochastic Model Predictive Control addresses uncertainties by incorporating chance constraints that provide probabilistic guarantees of constraint satisfaction. However, simultaneously optimizing over the risk allocation and the feedback…
Maximal inequalities refer to bounds on expected values of the supremum of averages of random variables over a collection. They play a crucial role in the study of non-parametric and high-dimensional estimators, and especially in the study…
We provide verification theorems (at different levels of generality) for infinite horizon stochastic control problems in continuous time for semimartingales. The control framework is given as an abstract "martingale formulation", which…
We obtain functional central limit theorems for both discrete time expressions of the form $1/\sqrt{N}\sum_{n=1}^{[Nt]}(F(X(q_1(n)),\ldots, X(q_{\ell}(n)))-\bar{F})$ and similar expressions in the continuous time where the sum is replaced…