Almost sure asymptotic stability of stochastic difference and differential equations with non-anticipating memory terms is studied in R. Sufficient criteria are obtained with help of Lyapunov-Krasovskǐi-type functionals, martingale decomposition and semi-martingale convergence theorems. The results allow numerical methods for stochastic differential equations with memory to be studied in terms ...

The time-reversibility of a Markov process implies a particular structure of the score function. It is explored which martingale estimating functions and other unbiased estimating functions have a similar structure. This leads to an estimating function with a semiparametric efficiency property. Also relations to martingale estimating functions based on eigenfunctions of the infinitesimal genera...

2 Binary search trees 2 2.1 Definition of a binary search tree . . . . . . . . . . . . . . . . . . 2 2.2 Profile. Discrete martingale . . . . . . . . . . . . . . . . . . . . . 3 2.3 Embedding in continuous time. Yule tree . . . . . . . . . . . . . . 4 2.4 Martingale connection Yule tree binary search tree . . . . . . . . 5 2.5 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

Weak uniqueness is established for the martingale problem associated to a family of catalytic branching networks. This martingale problem corresponds to a stochastic differential equation with a degenerate Hölder continuous diffusion matrix. Our approach uses the semigroup perturbation method of Stroock and Varadhan and a modification of a Banach space of weighted Hölder continuous functions in...

In this paper we study multivalued martingales in continuous time. First we show that every multivalued martingale in continuous time can be represented as the closure of a sequence of martingale selections. Then we prove two results concerning the cadlag modifications of continuous time multivalued martingales, in Kuratowski-Mosco convergence and in convergence in the Hausdorff metric respecti...

We prove the analogue of the classical Burkholder-Gundy inequalites for noncommutative martingales. As applications we give a characterization for an Ito-Clifford integral to be an L-martingale via its integrand, and then extend the Ito-Clifford integral theory in L, developed by Barnett, Streater and Wilde, to L for all 1 < p < ∞. We include an appendix on the non-commutative analogue of the c...

The original transport problem is to optimally move a pile of soil to an excavation. Mathematically, given two measures of equal mass, we look for an optimal map that takes one measure to the other one and also minimizes a given cost functional. Kantorovich relaxed this problem by considering a measure whose marginals agree with given two measures instead of a bijection. This generalization lin...

In recent work Long and Servedio [LS05] presented a “martingale boosting” algorithm that works by constructing a branching program over weak classifiers and has a simple analysis based on elementary properties of random walks. [LS05] showed that this martingale booster can tolerate random classification noise when it is run with a noise-tolerant weak learner; however, a drawback of the algorith...