Embedding of random vectors into continuous martingales

E. Dettweiler

Displaying similar documents to “Embedding of random vectors into continuous martingales”

A recurrence theorem for square-integrable martingales

Gerold Alsmeyer (1994)

Studia Mathematica

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Let ${(M_{n})}_{n \geq 0}$ be a zero-mean martingale with canonical filtration ${(ℱ_{n})}_{n \geq 0}$ and stochastically $L_{2}$ -bounded increments $Y_{1}, Y_{2}, . . .,$ which means that $P (| Y_{n} | > t | ℱ_{n - 1}) \leq 1 - H (t)$ a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let $V^{2} = \sum_{n \geq 1} E (Y_{n}^{2} | ℱ_{n - 1})$ . It is the main result of this paper that each such martingale is a.s. convergent on V < ∞ and recurrent on V = ∞, i.e. $P (M_{n} \in [- c, c] i . o . | V = \infty) = 1$ for some c > 0. This generalizes a recent result by Durrett, Kesten and Lawler [4] who consider the case of only finitely many square-integrable increment...

Weak convergence to Ocone martingales: a remark.

Peccati, Giovanni (2004)

Electronic Communications in Probability [electronic only]

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On the Lévy transformation of brownian motions and continuous martingales

Lester E. Dubins, Michel Émery, Marc Yor (1993)

Séminaire de probabilités de Strasbourg

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Continuous Ocone martingales as weak limits of rescaled martingales.

van Zanten, Harry (2002)

Electronic Communications in Probability [electronic only]

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Some invariance properties (of the laws) of Ocone's martingales

Lioudmilla Vostrikova, Marc Yor (2000)

Séminaire de probabilités de Strasbourg

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A sharp maximal inequality for continuous martingales and their differential subordinates

Adam Osękowski (2013)

Czechoslovak Mathematical Journal

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Assume that $X$ , $Y$ are continuous-path martingales taking values in $ℝ^{ν}$ , $ν \geq 1$ , such that $Y$ is differentially subordinate to $X$ . The paper contains the proof of the maximal inequality $∥ sup_{t \geq 0} | Y_{t} {| ∥}_{1} \leq 2 ∥ sup_{t \geq 0} | X_{t} {| ∥}_{1} .$ The constant $2$ is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder’s method and rests on the construction of an appropriate special function.

The Skorokhod embedding problem and its offspring.

Obłój, Jan (2004)

Probability Surveys [electronic only]

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Some remarkable pure martingales

S. Beghdadi-Sakrani (2003)

Annales de l'I.H.P. Probabilités et statistiques

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Hiding a constant drift

Vilmos Prokaj, Miklós Rásonyi, Walter Schachermayer (2011)

Annales de l'I.H.P. Probabilités et statistiques

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The following question is due to Marc Yor: Let be a brownian motion and =+ . Can we define an -predictable process such that the resulting stochastic integral (⋅) is a brownian motion (without drift) in its own filtration, i.e. an -brownian motion? In this paper we show that by dropping the requirement of -predictability of we can give a positive answer to this question. In other words, we are able to show that there is a weak solution to Yor’s question....