Convergence of Riesz space martingales

We prove a martingale convergence for sub and super martingales on Riesz spaces. As a consequence we can form Krickeberg and Riesz like decompositions. The minimality of the Krickeberg decomposition yields a natural ordered lattice structure on the space of convergent martingales making this space into a Dedekind complete Riesz space. Finally we show that the Riesz space of convergent martingales is Riesz isomorphic to the order closure of the union of the ranges of the conditional expectations in the filtration. Consequently we can characterize the space of order convergent martingales both in Riesz spaces and in the setting of probability spaces.     

Cramér large deviation expansions for martingales under Bernstein’s condition

An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramér to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramér result and therefore are optimal. Our result implies a moderate deviation principle for martingales.     

On the characterisation of honest times that avoid all stopping times

We present a short and self-contained proof of the following result: a random time is an honest time that avoids all stopping times if and only if it coincides with the (last) time of maximum of a nonnegative local martingale with zero terminal value and no jumps while at its running supremum, where the latter running supremum process is continuous. Illustrative examples involving local martingales with discontinuous paths are provided.     

Atomic decompositions of Banach lattice-valued martingales

In this paper, atomic decompositions of Banach lattice-valued martingales are given. We discuss the relation between the LERMT property and atomic decompositions. With the help of atomic decompositions, the relation of the martingale spaces is investigated.     

Strong Martingales: Their Decompositions and Quadratic Variation

Set-indexed strong martingales and a form of predictability for set-indexed processes are defined. Under a natural integrability condition, we show that any set-indexed strong submartingale can be decomposed in the Doob–Meyer sense. A form of predictable quadratic variation for square-integrable set-indexed strong martingales is defined and sufficient conditions for its existence are given. Under a conditional independence assumption, these reduce to a simple moment condition and, if the strong martingale has continuous sample paths, the resulting quadratic variation can be approximated in the L ₂-sense by sums of conditional expectations of squared increments.     

Differentiable and analytic families of continuous martingales in manifolds with connection

Summary.  We prove that the derivative of a differentiable family X _t (a) of continuous martingales in a manifold M is a martingale in the tangent space for the complete lift of the connection in M, provided that the derivative is bicontinuous in t and a. We consider a filtered probability space (Ω,(ℱ _t )_{0≤ t ≤1}, ℙ) such that all the real martingales have a continuous version, and a manifold M endowed with an analytic connection and such that the complexification of M has strong convex geometry. We prove that, given an analytic family a↦L(a) of random variable with values in M and such that L(0)≡x ₀∈M, there exists an analytic family a↦X(a) of continuous martingales such that X ₁(a)=L(a). For this, we investigate the convexity of the tangent spaces T ^{( n )} M, and we prove that any continuous martingale in any manifold can be uniformly approximated by a discrete martingale up to a stopping time T such that ℙ(T<1) is arbitrarily small. We use this construction of families of martingales in complex analytic manifolds to prove that every ℱ₁-measurable random variable with values in a compact convex set V with convex geometry in a manifold with a C ¹ connection is reachable by a V-valued martingale. Received: 14 March 1996/In revised form: 12 November 1996     

Exit probability estimates for martingales in geodesic balls,using curvature

Summary Kallenberg and Sztencel have recently discovered exponential upper bounds, independent of dimension, on the probability that a vector martingale will exit from a ball in Euclidean space by timet. This article extends their results to martingales on Riemannian manifolds, including Brownian motion, and shows how exit probabilities depend on curvature. Using comparison with rotationally symmetric manifolds, these estimates are easily computable, and are sharp up to a constant factor in certain cases.     

Regularity and decomposition of two-parameter supermartingales

The well-known Doob-Meyer decomposition of a supermartingale as the difference of a martingale and an increasing process is extended in several ways for two-parameter stochastic processes. In particular, the notion of laplacian is introduced which gives more explicit decomposition for potentials. The optional sampling theorem is stated for a wide class of supermartingales justifying the study of local martingales. Conditions for regularity and continuity for two-parameter processes are given using approximate laplacians. By introducing the notion of optional increasing path, the relation between the regularity of certain quasimartingales and the continuity of the associated integrable variation process is proved.     

Stochastic Integrals of Continuous Local Martingales,II

Consider a continuous local martingale X. We say that X satisfies the representation property if any martingale Y of X can be represented as stochastic ITÔ integral of X. On the basis of part I of the present paper, in section 4 several general examples of continuous local martingales X satisfying the representation property are given: Stochastic continuous GAUSSian martingales, processes with conditionally independent increments, stopped continuous local martingales, random time change of WIENER processes, weak solutions of stochastic differential equations. Theorem 7 states that every (homogeneous) continuous strong MARKOV local martingale has the representation property. In section 5, the results of part I are applied to n-dimensional continuous local martingales and analogous representation results are obtained. In section 6, we consider an application of section 5 to the n-dimensional time change for reducing every n-dimensional continuous local martingale with orthogonal components to the WIENER process. This improves a theorem of F. B. KNIGHT and simplifies its proof considerably.     

Probability set functions

A probability set function is interpretable as a probability distribution on binary sequences of fixed length. Cumulants of probability set functions enjoy particularly simple properties which make them more manageable than cumulants of general random variables. We derive some identities satisfied by cumulants of probability set functions which we believe to be new. Probability set functions may be expanded in terms of their cumulants. We derive an expansion which allows the construction of examples of probability set functions whose cumulants are arbitrary, restricted only by their absolute values. It is known that this phenomenon cannot occur for continuous probability distributions. Some particular examples of probability set functions are considered, and their cumulants are computed, leading to a conjecture on the upper bound of the values of cumulants. Moments of probability set functions determined by arithmetical conditions are computed in a final example.Dedicated to our friend, W.A. Beyer. Financial support for this work was derived from the U.S.D.O.E. Human Genome Project, through the Center for Human Genome Studies at Los Alamos National Laboratory, and also through the Center for Nonlinear Studies, Los Alamos National Laboratory, LANL report LAUR-97-323.     

Duality theorem for B‐valued martingale Orlicz–Hardy spaces associated with concave functions

The dual space of B ‐valued martingale Orlicz–Hardy space with a concave function Φ, which is associated with the conditional p‐variation of B ‐valued martingale, is characterized. To obtain the results, a new type of Campanato spaces for B ‐valued martingales is introduced and the classical technique of atomic decompositions is improved. Some results obtained here are connected closely with the p‐uniform smoothness and q‐uniform convexity of the underlying Banach space.     

Stochastic integrals in Riemann manifolds

Stochastic integrals are constructed with values in a compact Riemann manifold from a continuous martingale integrator that is given in the tangent space of the initial point of the stochastic integral and from a stochastic tensor field of linear endomorphisms of the tangent bundle. The integrals that are formed are continuous processes that suitably preserve the martingale property. These stochastic integrals should be useful for the applications of a stochastic calculus in Riemann manifolds.     

Exponential integrability of martingales

We obtain the exponential integrability of the maximal function, the quadratic variation and the conditional quadratic variation of bounded martingales and exponential integrable martingales.     

A canonical setting and separating times for continuous local martingales

The notion of a separating time for a pair of measures on a filtered space is helpful for studying problems of (local) absolute continuity and singularity of measures. In this paper, we describe a certain canonical setting for continuous local martingales (abbreviated below as CLMs) and find an explicit form of separating times for CLMs in this setting.     

Brownian motion and the formation of singularities in the heat flow for harmonic maps

Summary We develop a general framework for a stochastic interpretation of certain nonlinear PDEs on manifolds. The linear operation of takin expectations is replaced by the concept of martingale means, namely the notion of deterministic starting points of martingales (with respect to the Levi-Civita connection) ending up at a prescribed state. We formulate a monotonicity condition for the Riemannian quadratic variation of such martingales that allows us to turn smallness of the quadratic variation into a priori gradient bounds for solutions of the nonlinear heat equation. Such estimates lead to simple criteria for blow-ups in the nonlinear heat flow for harmonic maps with small initial energy.This article was processed by the author using the Springer-Verlag TEX QPMZGHB macro package 1991.     

Weak martingale Hardy spaces and weak atomic decompositions

In this paper we define some weak martingale Hardy spaces and three kinds of weak atoms. They are the counterparts of martingale Hardy spaces and atoms in the classical martingale Hp-theory. And then three atomic decomposition theorems for martingales in weak martingale Hardy spaces are proved. With the help of the weak atomic decompositions of martingale, a sufficient condition for a sublinear operator defined on the weak martingale Hardy spaces to be bounded is given. Using the sufficient condition, we obtain a series of martingale inequalities with respect to the weak Lp-norm, the inequalities of weak (p ,p)-type and some continuous imbedding relationships between various weak martingale Hardy spaces. These inequalities are the weak versions of the basic inequalities in the classical martingale Hp-theory.     

A martingale characterization of the set-indexed Brownian motion

We characterize a Brownian motion indexed by a semilattice of sets, using the theory of set-indexed martingales: a square integrable continuous set-indexed strong martingale is a Brownian motion if and only if its compensator is deterministic and continuous.Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada.Research done while this author was visiting the University of Ottawa. He wishes to thank Professor Ivanoff for her kind hospitality.     

On the conditioned exit measures of super Brownian motion

In this paper we present a martingale related to the exit measures of super Brownian motion. By changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain. The results are similar in flavor to the “immortal particle” picture of conditioned super Brownian motion but more general, as the change of measure is given by a martingale which need not arise from a single harmonic function. Received: 27 August 1998 / Revised version: 8 January 1999     

Central Limit Theorems for approximate quadratic variations of pure jump Itô semimartingales

We derive Central Limit Theorems for the convergence of approximate quadratic variations, computed on the basis of regularly spaced observation times of the underlying process, toward the true quadratic variation. This problem was solved in the case of an Itô semimartingale having a non-vanishing continuous martingale part. Here we focus on the case where the continuous martingale part vanishes and find faster rates of convergence, as well as very different limiting processes.