Related papers: Doob's optional sampling and maximal inequality fo…
The Doob-Dynkin Lemma gives conditions on two functions $X$ and $Y$ that ensure existence of a function ${\phi}$ so that $X = {\phi} \circ Y$. This communication proves different versions of the Doob-Dynkin Lemma, and shows how it is…
In this paper we consider a method of solving optimal stopping problems in discrete and continuous time based on their dual representation. A novel and generic simulation-based optimization algorithm not involving nested simulations is…
We consider a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, which is endowed with two filtrations, $\mathbb{G}$ and $\mathbb{F}$, assumed to satisfy the usual conditions and such that $\mathbb{F} \subset \mathbb{G}$. On this…
The purpose of this paper is to give a survey of a class of maximal inequalities for purely discontinuous martingales, as well as for stochastic integral and convolutions with respect to Poisson measures, in infinite dimensional spaces.…
In this paper, we develop necessary and sufficient conditions for the validity of a martingale approximation for the partial sums of a stationary process in terms of the maximum of consecutive errors. Such an approximation is useful for…
The martingale expansion provides a refined approximation to the marginal distributions of martingales beyond the normal approximation implied by the martingale central limit theorem. We develop a martingale expansion framework specifically…
We present a new deep primal-dual backward stochastic differential equation framework based on stopping time iteration to solve optimal stopping problems. A novel loss function is proposed to learn the conditional expectation, which…
We study the existing algorithms that solve the multidimensional martingale optimal transport. Then we provide a new algorithm based on entropic regularization and Newton's method. Then we provide theoretical convergence rate results and we…
We propose \textit{DeepMartingale}, a deep-learning framework for the dual formulation of discrete-monitoring optimal stopping problems under continuous-time models. Leveraging a martingale representation, our method implements a…
In this paper, we consider a class of stochastic optimal control problems with risk constraints that are expressed as bounded probabilities of failure for particular initial states. We present here a martingale approach that diffuses a risk…
Sampling theory concerns the problem of reconstruction of functions from the knowledge of their values at some discrete set of points. In this paper we derive an orthogonal sampling theory and associated Lagrange interpolation formulae from…
Rare event sampling in dynamical systems is a fundamental problem arising in the natural sciences, which poses significant computational challenges due to an exponentially large space of trajectories. For settings where the dynamical system…
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…
Gibbs random fields play an important role in statistics, for example the autologistic model is commonly used to model the spatial distribution of binary variables defined on a lattice. However they are complicated to work with due to an…
We give sufficient conditions for the bounded law of the iterated logarithms for strictly stationary random fields when the summation is done on rectangle. The study is done by the control of an appropriated maximal function. The case of…
Recently surrogate functions based on the tail inequalities were developed to evaluate the chance constraints in the context of evolutionary computation and several Pareto optimization algorithms using these surrogates were successfully…
A stationary random sequence admits under some assumptions a representation as the sum of two others: one of them is a martingale difference sequence, and another is a so-called coboundary. Such a representation can be used for proving some…
Motivated by recent results on the dual formulation of optimal stopping problems, we investigate in this short paper how the knowledge of an approximating dual martingale can improve the efficiency of primal methods. In particular, we show…
Evaluation of the Bellman functions is a difficult task. The exact Bellman functions of the dyadic Carleson Embedding Theorem 1.1 and the dyadic maximal operators are obtained in [3] and [4]. Actually, the same Bellman functions also work…
We prove martingale-ergodic and ergodic-martingale theorems for vector valued Bochner integrable functions. We obtain dominant and maximal inequalities. We also prove weighted and multiparameter martingale-ergodic and ergodic martingale…