Planar semi-martingales

Semi-martingales and rough paths theory

Abstract: We prove that the theory of rough paths, which is used to define path-wise integrals and path-wise differential equations, can be used with continuous semi-martingales. We provide then an almost sure theorem of type Wong-Zakai. Moreover, we show that the conditions UT and UCV, used to prove that one can interchange limits and Itô or Stratonovich integrals, provide the same result when...

Concentration Inequalities for Semi-bounded Martingales

In this paper we extend the results of de la Peña [3]. The main method that we use is the theory of decoupling, which has been developed in de la Peña [2] and [3]. Decoupling theory provides a general framework for analyzing problems involving dependent random variables as if they were independent. We will apply the theory of decoupling as in de la Peña [3] and some new inequalities for indepen...

Local Times and Almost Sure Convergence of Semi-Martingales

Semi-Planar Laser Holographic Display

Ergophos LLC has developed and tested a new and unique class of illumination devices for mobile LCD displays (including autostereo), microscopy, and medical instruments. The devices are based on semi-planar holograms which deliver laser light with unprecedented accuracy and can be mass produced with conventional technology. The energy efficiency of a laser holographic display panel is a factor ...

Continuous Martingales and Local Martingales

Throughout these notes, (Ω,F ,P ) will be a probability space and F := {Ft }t∈J a filtration indexed by J , where J is an interval, usually J = [0,∞). The filtration F is said to be complete if each Ft contains all sets of measure 0, and is right-continuous if Ft =∩s>t Fs . A standard filtration is a filtration that is both complete and right-continuous. A stochastic process {X t }t≥0 defined o...