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.     

Efficiency analysis, shortage functions, arbitrage, and martingales

This paper shows that standard tools of efficiency analysis, directional distance functions, can be used to characterize the investment-returns technology. That ability to characterize the investment-returns technology and fundamental duality relationships imply that directional distance functions can be used to detect the presence of an arbitrage, to value financial assets in the absence of an arbitrage lying in the span of the market and to place bounds on the no-arbitrage values of assets lying outside the span of the market.     

On Schnorr and computable randomness,martingales, and machines

We examine the randomness and triviality of reals using notions arising from martingales and prefix‐free machines. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Functional equations and martingales

Aequationes mathematicae - We consider functional equations (Cauchy’s, Abel’s and some other functional equations) and show that finding the general solution of these equations...     

Hamming cube and martingales

In the present paper, we show that a correctly chosen Legendre transform of the Bellman functions of martingale problems give us the right tool to prove isoperimetric inequalities on Hamming cube independent of the dimension. We illustrate the power of this “dual function approach” by proving certain Poincaré inequalities on Hamming cube and by improving a particular inequality of Beckner on the Hamming cube.     

Fermion martingales

Summary We show that strictly quasi-free Fermion martingales may be expressed as a sum of quantum stochastic integrals with respect to the Fermi creation and annihilation processes and a multiple of the identity.     

Integrals,conditional expectations,and martingales of multivalued functions

Let (Ω, $\text{A}$, μ) be a finite measure space and $\text{X}$ a real separable Banach space. Measurability and integrability are defined for multivalued functions on Ω with values in the family of nonempty closed subsets of $\text{X}$. To present a theory of integrals, conditional expectations, and martingales of multivalued functions, several types of spaces of integrably bounded multivalued functions are formulated as complete metric spaces including the space L¹(Ω; $\text{X}$) isometrically. For multivalued functions in these spaces, multivalued conditional expectations are introduced, and the properties possessed by the usual conditional expectation are obtained for the multivalued conditional expectation with some modifications. Multivalued martingales are also defined, and their convergence theorems are established in several ways.     

The critical random graph,with martingales

We give a short proof that the largest component C ₁ of the random graph G(n, 1/n) is of size approximately n ^2/3. The proof gives explicit bounds for the probability that the ratio is very large or very small. In particular, the probability that n ^−2/3|C ₁| exceeds A is at most e ^{- cA3}{e^{ - c{A^3}}} for some c > 0.     

Measure free martingales

We give a necessary and sufficient condition on a sequence of functions on a set Ω under which there is a measure on Ω which renders the given sequence of functions a martingale. Further such a measure is unique if we impose a natural maximum entropy condition on the conditional probabilities.     

Complex martingales and asymptotic enumeration

Many enumeration problems in combinatorics, including such fundamental questions as the number of regular graphs, can be expressed as high‐dimensional complex integrals. Motivated by the need for a systematic study of the asymptotic behavior of such integrals, we establish explicit bounds on the exponentials of complex martingales. Those bounds applied to the case of truncated normal distributions are precise enough to include and extend many enumerative results of Barvinok, Canfield, Gao, Greenhill, Hartigan, Isaev, McKay, Wang, Wormald, and others. Our method applies to sums as well as integrals. As a first illustration of the power of our theory, we considerably strengthen existing results on the relationship between random graphs or bipartite graphs with specified degrees and the so‐called β‐model of random graphs with independent edges, which is equivalent to the Rasch model in the bipartite case.     

Two-parameter diffusion processes and martingales

We introduce a class of two-parameter processes which are diffusions on each coordinate and satisfy a particular Markov property related to the partial ordering in R²₊. These processes can be expressed as solutions of some stochastic integral equations driven by a two-parameter Wiener process and two families of ordinary Brownian motions. This result is based on a characterization of two-parameter martingales with orthogonal increments.     

On martingales and feller semigroups

Let E be a second countable locally compact Hausdorff space and let L be a linear operator with domain D(L) and range R(L) in C0(E). Suppose that D(L) is dense in E. The following assertions are equivalent:

1. For L the martingale problem is uniquely solvable and L is maximal for this property
2. The operator L generates a Feller semigroup in C0(E).

     

Quasimartingales,martingales locales,semimartingales et filtration naturelle

Résumé L'objet de cet article est de démontrer que certaines propriétés d'un processus stochastique sont préservées par restriction de la famille de tribus à la filtration naturelle du processus. Le résultat principal dit que toute semimartingale reste une semimartingale pour sa filtration naturelle, mais pour établir cela nous traitons le cas des quasimartingales, et des martingales locales bornées dansL ¹.