It is shown that any nonnegative bounded supermartingale admits a Doob-Meyer decomposition as a difference of a martingale and an adapted increasing process upon appropriate choice of a reference probability measure which may be only finitely additive. Introduction. In [Armstrong, 1983] it is shown that every bounded finitely additive supermartingale is a decreasing process with respect to some...

Pitman's theorem states that if {Bt, t ≥ 0} is a one-dimensional Brownian motion, then {Bt − 2 inf s≤t Bs, three dimensional Bessel process, i.e. motion conditioned in Doob sense to remain forever positive. In this paper one gives similar representation for the an interval. Due double barrier, it much more involved and only asymptotic. This uses fact interval alcove of Affine Lie algebra A 1 .

Given a right continuous family Ft of complete o-fields and a bounded right continuous family Xt of random variables, we show in this paper that it is possible to modify the conditional expectations E(Xt\Ft) to be right continuous. When Xt=X, this reduces to a result of J. L. Doob. A famous result of Doob states that any martingale has a right continuous with left limits modification. This is a...

This paper aims to provide some tools coming from functional inequalities deal with quasi-stationarity for absorbed Markov processes. First, it is shown how a Poincaré inequality related suitable Doob transform entails exponential convergence of conditioned distributions quasi-stationary distribution in total variation and 1-Wasserstein distance. A special attention paid multi-dimensional diffu...

This article examines neural network-based approximations for the superhedging price process of a contingent claim in discrete time market model. First we prove that α-quantile hedging converges to at 0 α tending 1, and show can be approximated by price. provides approximation also strategy up maturity. To obtain t > $t>0$ , using Doob decomposition, it is sufficient determine consumption. We e...

Stimulated by a remark of J. L. Doob at the beginning of Appendix I to Kai Lai Chung’s English translation “Limit Distributions for Sums of Independent Variables” of the Russian classic by B. V. Gnedenko and A. N. Kolmogorov [5] we highlight the somewhat nonprobabilistic importance of characteristic functions and their positive definiteness property in a pedagogical attempt to introduce the not...

In this work, a class of multidimensional stochastic hybrid dynamic models is studied. The system under investigation is a first-order linear nonhomogeneous system of Itô-Doob type stochastic differential equations with switching coefficients. The switching of the system is governed by a discrete dynamic which is monitored by a non-homogeneous Poisson process. Closed-form solutions of the syste...

During the three decades from 1930 to 1960 J. L. Doob was, with the possible exception of Kolmogorov, the man most responsible for the transformation of the study of probability to a mathematical discipline. His accomplishments were recognized by both probabilists and other mathematicians in that he is the only person ever elected to serve as president of both the IMS and the AMS. This article ...