Mean Convergence of Martingales

Uniform Convergence of Reversed Martingales

A necessary and sufficient condition for the uniform convergence of a family of reversed martingales converging to a degenerated limiting process is given. The condition is expressed by means of regular convergence (in Hardy’s sense) of corresponding means. It is shown that the given regular convergence is equivalent to Hoffmann-Jørgensen’s eventually total boundedness in the mean which is nece...

On The Mean Convergence of Biharmonic Functions

Let denote the unit circle in the complex plane. Given a function , one uses t usual (harmonic) Poisson kernel for the unit disk to define the Poisson integral of , namely . Here we consider the biharmonic Poisson kernel for the unit disk to define the notion of -integral of a given function ; this associated biharmonic function will be denoted by . We then consider the dilations ...

Mean square convergence of a semidiscrete scheme for SPDEs of Zakai type driven by square integrable martingales

In this short note, a direct proof of L2 convergence of an Euler–Maruyama approximation of a Zakai equation driven by a square integrable martingale is shown. The order of convergence is as known for real-valued stochastic differential equations and for less general driving noises O( √ ∆t) for a time discretization step size ∆t.

Examples and counterexamples to almost-sure convergence of bilateral martingales

Given a stationary process (Xp)p∈Z and an event B ∈ σ(Xp, p ∈ Z), we study the almost sure convergence as n and m go to infinity of the “bilateral” martingale E [1B |X−n, X−n+1, . . . , Xm−1, Xm ] . We show that almost sure convergence holds in some classical examples such as i.i.d. or Markov processes, as well as for the natural generator of Chacon’s transformation. However, we also prove that...

Local Times and Almost Sure Convergence of Semi-Martingales