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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 0 be a zero-mean martingale with canonical filtration ( n ) n 0 and stochastically L 2 -bounded increments Y 1 , Y 2 , . . . , which means that P ( | Y n | > t | n - 1 ) 1 - H ( t ) a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let V 2 = n 1 E ( Y n 2 | 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 [ - c , c ] i . o . | V = ) = 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...

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 ν , ν 1 , such that Y is differentially subordinate to X . The paper contains the proof of the maximal inequality sup t 0 | Y t | 1 2 sup t 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.

Hiding a constant drift

Vilmos Prokaj, Miklós Rásonyi, Walter Schachermayer (2011)

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

Similarity:

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....