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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 3, Fasc. 1,
pages 97 - 101
 

ON THE ALMOST SURE CONVERGENCE OF THE SQUARE VARIATION OF THE BROWNIAN MOTION

Andrzej Wróbel

Abstract: The paper deals with the problem of almost sure (a.s.) convergence of the square variation of the Brownian motion when the diameters d n of partitions of the time interval tend to zero. It is known that if the diameters converge fast enough, namely if d n is of order less than lg-1n , then a.s. convergence takes place. On the other hand, we show that there exists a sequence of partitions with diameters d n of order less than lg-an for any 0 < a < 1 such that the Brownian square variation diverges a.s.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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