Bounded Martingale

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Bernard De Meyer – 1st expert on this subject based on the ideXlab platform

• Price Dynamics on a Stock Market with Asymmetric Information
Levine's Bibliography, 2020
Co-Authors: Bernard De Meyer

Abstract:

When two asymmetrically informed risk-neutral agents repeatedly exchange a risky asset for numeraire, they are essentially playing an n-times repeated zero-sum game of incomplete information. In this setting, the price Lq at period q can be defined as the expected liquidation value of the risky asset given players’ past moves. This paper indicates that the asymptotics of this price process at equilibrium, as n goes to [infinity], is completely independent of the “natural” trading mechanism used at each round: it converges, as n increases, to a Continuous Martingale of Maximal Variation. This Martingale class thus provides natural dynamics that could be used in financial econometrics. It contains in particular Black and Scholes’ dynamics. We also prove here a mathematical theorem on the asymptotics of Martingales of maximal M-variation, extending Mertens and Zamir’s paper on the maximal L1-variation of a Bounded Martingale. (This abstract was borrowed from another version of this item.)

• Price dynamics on a stock market with asymmetric information
Games and Economic Behavior, 2010
Co-Authors: Bernard De Meyer

Abstract:

When two asymmetrically informed risk-neutral agents repeatedly exchange a risky asset for numeraire, they are essentially playing an n-times repeated zero-sum game of incomplete information. In this setting, the price Lq at period q can be defined as the expected liquidation value of the risky asset given players’ past moves. This paper indicates that the asymptotics of this price process at equilibrium, as n goes to [infinity], is completely independent of the “natural” trading mechanism used at each round: it converges, as n increases, to a Continuous Martingale of Maximal Variation. This Martingale class thus provides natural dynamics that could be used in financial econometrics. It contains in particular Black and Scholes’ dynamics. We also prove here a mathematical theorem on the asymptotics of Martingales of maximal M-variation, extending Mertens and Zamir’s paper on the maximal L1-variation of a Bounded Martingale.

• the maximal variation of a BoundedMartingale and the central limit theorem
Annales De L Institut Henri Poincare-probabilites Et Statistiques, 1998
Co-Authors: Bernard De Meyer

Abstract:

Mertens and Zamir’s (1977) paper is concerned with the asymptotic behaviour of the maximal L[exp.1]-variation [xi1.n(p)] of a [0,1]-valued Martingale of length n starting at p. They prove the convergence of [ [xi1.n(p)] / [square root.n]]. to the normal density evaluated at its p-quantile. This paper generalises this result to the conditional L[exp.q]-variation for q [belong] [1,2). The appearance of the normal density remained unexplained in Mertens and Zamir’s proof: it appeared there as the solution of a differential equation. Our proof however justifies this normal density as a consequence of a generalisation of the CLT discussed in the second part of this paper.

François Perron – 2nd expert on this subject based on the ideXlab platform

• A New Randomized Pólya Urn Model
Applied Mathematics, 2020
Co-Authors: Djilali Ait Aoudia, François Perron

Abstract:

<span>In this paper, we propose a new class of discrete time stochastic processes generated by a two-color generalized Pólya urn, that is reinforced every time. A single urn contains </span><i><span>a</span></i><span> white balls, </span><i><span>b</span></i><span> black balls and evolves as follows: at discrete times </span><i><span>n</span></i><span>=1,2,…,</span><span> we sample</span><i><span> M<sub>n </sub></span></i><span>balls and note their colors, say</span><i><span> R<sub>n </sub></span></i><span>are white and </span><i><span>M<sub>n</sub>- R<sub>n</sub></span></i><span> are black. We return the drawn balls in the urn. Moreover, </span><i><span>N<sub>n</sub>R<sub>n </sub></span></i><span>new white balls and </span><i><span>N<sub>n</sub></span></i><span> (</span><i><span>M<sub>n</sub>- R<sub>n</sub></span></i><span>)</span><i><span> </span></i><span>new black balls are added in the urn. The numbers</span><i><span> M<sub>n </sub></span></i><span>and</span><i><span> N<sub>n </sub></span></i><span>are random variables. We show that the proportions of white balls forms a Bounded Martingale sequence which converges almost surely. Necessary and sufficient conditions for the limit to concentrate on the set </span><span>{0,1} </span><span>are given.</span><span></span&gt

• A New Randomized Pólya Urn Model
Applied Mathematics-a Journal of Chinese Universities Series B, 2012
Co-Authors: Djilali Ait Aoudia, François Perron

Abstract:

In this paper, we propose a new class of discrete time stochastic processes generated by a two-color generalized Polya urn, that is reinforced every time. A single urn contains a white balls, b black balls and evolves as follows: at discrete times n=1,2,…, we sample Mn balls and note their colors, say Rn are white and Mn- Rn are black. We return the drawn balls in the urn. Moreover, NnRn new white balls and Nn (Mn- Rn) new black balls are added in the urn. The numbers Mn and Nn are random variables. We show that the proportions of white balls forms a Bounded Martingale sequence which converges almost surely. Necessary and sufficient conditions for the limit to concentrate on the set {0,1} are given.

Ashkan Nikeghbali – 3rd expert on this subject based on the ideXlab platform

• A definition and some characteristic properties of pseudo-stopping times
Annals of Probability, 2005
Co-Authors: Ashkan Nikeghbali

Abstract:

Recently, Williams [Bull. London Math. Soc. 34 (2002) 610-612] gave an explicit example of a random time p associated with Brownian motion such that p is not a stopping time but EMp = EM 0 for every Bounded Martingale M. The aim of this paper is to characterize such random times, which we call pseudo-stopping times, and to construct further examples, using techniques of progressive enlargements of filtrations.

• A definition and some characteristic properties of pseudo-stopping times
arXiv: Probability, 2004
Co-Authors: Ashkan Nikeghbali

Abstract:

Recently, D. Williams \cite{williams} gave an explicit example of a random time $\rho$ associated with Brownian motion such that $\rho$ is not a stopping time but $\mathbb{E}M_{\rho}=\mathbb{E}M_{0}$ for every Bounded Martingale $M$. The aim of this paper is to give some characterizations for such random times, which we call pseudo-stopping times, and to construct further examples, using techniques of progressive enlargements of filtrations.