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Bernard De Meyer - One of the best experts 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 Bounded Martingale 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 - One of the best experts 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>

  • 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 - One of the best experts 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.

Abdelkarem Berkaoui - One of the best experts on this subject based on the ideXlab platform.

Isaac Meilijson - One of the best experts on this subject based on the ideXlab platform.

  • a sharp bound on the expected local time of a continuous cal l _2 Bounded Martingale
    arXiv: Probability, 2020
    Co-Authors: David Gilat, Isaac Meilijson, Laura Sacerdote
    Abstract:

    For a continuous ${\cal L}_2$-Bounded Martingale with no intervals of constancy, starting at $0$ and having final variance $\sigma^2$, the expected local time at $x \in \cal{R}$ is at most $\sqrt{\sigma^2+x^2}-|x|$. This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval $(x-\sqrt{\sigma^2+x^2},x+\sqrt{\sigma^2+x^2})$. Sharp bounds for the expected maximum, maximal absolute value, maximal diameter and maximal number of upcrossings of intervals, have been established by Dubins and Schwarz (1988), Dubins, Gilat and Meilijson (2009) and by the authors (2017).

  • a sharp bound on the expected number of upcrossings of an l2 Bounded Martingale
    Stochastic Processes and their Applications, 2017
    Co-Authors: David Gilat, Isaac Meilijson, Laura Sacerdote
    Abstract:

    Abstract For a Martingale M starting at x with final variance σ 2 , and an interval ( a , b ) , let Δ = b − a σ be the normalized length of the interval and let δ = | x − a | σ be the normalized distance from the initial point to the lower endpoint of the interval. The expected number of upcrossings of ( a , b ) by M is at most 1 + δ 2 − δ 2 Δ if Δ 2 ≤ 1 + δ 2 and at most 1 1 + ( Δ + δ ) 2 otherwise. Both bounds are sharp, attained by Standard Brownian Motion stopped at appropriate stopping times. Both bounds also attain the Doob upper bound on the expected number of upcrossings of ( a , b ) for subMartingales with the corresponding final distribution. Each of these two bounds is at most σ 2 ( b − a ) , with equality in the first bound for δ = 0 . The upper bound σ 2 on the length covered by M during upcrossings of an interval restricts the possible variability of a Martingale in terms of its final variance. This is in the same spirit as the Dubins & Schwarz sharp upper bound σ on the expected maximum of M above x , the Dubins & Schwarz sharp upper bound σ 2 on the expected maximal distance of M from x , and the Dubins, Gilat & Meilijson sharp upper bound σ 3 on the expected diameter of M .

  • on the expected diameter of an l2 Bounded Martingale
    Annals of Probability, 2009
    Co-Authors: Lester E Dubins, David Gilat, Isaac Meilijson
    Abstract:

    Dedicated to the memory of Gideon Schwarz (1933-2007) It is shown that the ratio between the expected diameter of an L2-Bounded Martingale and the standard deviation of its last term cannot exceed p 3. Moreover, a one-parameter family of stopping times on standard Brownian Motion is exhibited, for which the p 3 upper bound is attained. These stopping times, one for each cost-rate c, are optimal when the payoff for stopping at time t is the diameter D(t) obtained up to time t minus the hitherto accumulated cost ct. A quantity related to diameter, maximal drawdown (or rise), is introduced and its expectation is shown to be Bounded by p 2 times the standard deviation of the last term of the Martingale. These results complement the Dubins & Schwarz respective bounds 1 and p 2 for the ratios between the expected maximum and maximal absolute value of the Martingale and the standard deviation of its last term. Dynamic programming (gambling theory) methods are used for the proof of optimality.

  • on the expected diameter of an l2 Bounded Martingale
    arXiv: Probability, 2008
    Co-Authors: Lester E Dubins, David Gilat, Isaac Meilijson
    Abstract:

    It is shown that the ratio between the expected diameter of an L2-Bounded Martingale and the standard deviation of its last term cannot exceed sqrt(3). Moreover, a one-parameter family of stopping times on standard Brownian Motion is exhibited, for which the sqrt(3) upper bound is attained. These stopping times, one for each cost-rate c, are optimal when the payoff for stopping at time t is the diameter D(t) obtained up to time t minus the hitherto accumulated cost c t. A quantity related to diameter, maximal drawdown (or rise), is introduced and its expectation is shown to be Bounded by sqrt(2) times the standard deviation of the last term of the Martingale. These results complement the Dubins and Schwarz respective bounds 1 and sqrt(2) for the ratios between the expected maximum and maximal absolute value of the Martingale and the standard deviation of its last term. Dynamic programming (gambling theory) methods are used for the proof of optimality.