The Annals of Probability

Almost Sure Convergence of the Quadratic Variation of Martingales: A Counterexample

Itrel Monroe

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Abstract

Let $X_s$ be a continuous martingale and $Q\nu$ be an increasing sequence of partitions of [0, 1]. Let $$S^2(Q_\nu) = \sum_{t_i\in Q_\nu} (X_{t_i} - X_{t_{i - 1}})^2.$$ An example is given in which $$\lim \sup_{\nu \rightarrow \infty} S^2(Q_\nu) = \infty.$$

Article information

Source
Ann. Probab. Volume 4, Number 1 (1976), 133-138.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996192

Digital Object Identifier
doi:10.1214/aop/1176996192

Mathematical Reviews number (MathSciNet)
MR400384

Zentralblatt MATH identifier
0336.60048

JSTOR
links.jstor.org

Subjects
Primary: 60G45
Secondary: 60J65: Brownian motion [See also 58J65]

Keywords
Martingales quadratic variation square variation

Citation

Monroe, Itrel. Almost Sure Convergence of the Quadratic Variation of Martingales: A Counterexample. Ann. Probab. 4 (1976), no. 1, 133--138. doi:10.1214/aop/1176996192. https://projecteuclid.org/euclid.aop/1176996192


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