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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Brownian space-time functions of zero
quadratic variation depend only on time


Author: P. J. Fitzsimmons
Journal: Proc. Amer. Math. Soc. 127 (1999), 2423-2429
MSC (1991): Primary 60J65; Secondary 60J55
DOI: https://doi.org/10.1090/S0002-9939-99-04794-2
Published electronically: March 23, 1999
MathSciNet review: 1487367
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $B_ t$, $t\ge 0$, be a $1$-dimensional Brownian motion and let $f\colon {\mathbb{R}}\times [0,\infty [\,\to {\mathbb{R}}$ be a continuous function. We show that if $t\mapsto f(B_ t,t)$ is locally of zero quadratic variation, then $f(x,t)=f(0,t)$ for all $(x,t)\in {\mathbb{R}}\times [0,\infty [$. This result extends recent work of F. B. Knight, thereby confirming a conjecture of T.Salisbury.


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Additional Information

P. J. Fitzsimmons
Affiliation: Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093–0112
Email: pfitz@euclid.ucsd.edu

DOI: https://doi.org/10.1090/S0002-9939-99-04794-2
Keywords: Brownian motion, quadratic variation, time reversal, Girsanov transformation
Received by editor(s): September 8, 1997
Received by editor(s) in revised form: October 24, 1997
Published electronically: March 23, 1999
Communicated by: Stanley Sawyer
Article copyright: © Copyright 1999 American Mathematical Society