Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Published electronically: March 23, 1999
MathSciNet review: 1487367
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • [1] P. J. Fitzsimmons, Even and odd continuous additive functionals, Dirichlet Forms and Stochastic Processes (Z. M. Ma, M. Röckner, J. A. Yan, eds.), (Proc. Internat. Conf. on Dirichlet Forms and Stochastic Processes, Beijing, 1993, de Gruyter, Berlin, 1995, pp. 139-154. MR 97c:60191
  • [2] P. J. Fitzsimmons, Drift transformations of symmetric diffusions and duality, Preprint, 1997.
  • [3] H. Föllmer, P. Protter and A. N. Shiryaev, Quadratic variation and an extension of Itô's formula, Bernoulli 1 (1995), 149-169. MR 96k:60121
  • [4] K. Itô and H. P. McKean, Jr., Diffusions Processes and their Sample Paths, Second printing, corrected, Springer-Verlag, Berlin-Heidelberg-New York, 1974. MR 49:9963
  • [5] F. B. Knight, On a Brownian motion problem of T. Salisbury, Canad. Math. Bull. 40 (1997), 67-71. CMP 97:11
  • [6] D. Revuz, D. and M. Yor, Continuous Martingales and Brownian Motion, Second edition, Springer-Verlag, Berlin, 1994. MR 95h:60072
  • [7] T. S. Salisbury, An increasing diffusion, Seminar on Stochastic Processes, 1984 (E. Çinlar, K. L. Chung, R. K. Getoor, eds.), Birkhäuser, Boston, 1986, pp. 173-194. MR 88k:60138
  • [8] M. J. Sharpe, General Theory of Markov Processes, Academic Press, San Diego, 1988. MR 89m:60169
  • [9] H. Tanaka, Note on continuous additive functionals of the $1$-dimensional Brownian path, Z. Wahrscheinlichkeitstheorie und verw.Gebiete 1 (1962/63), 251-257. MR 29:6559
  • [10] J. B. Walsh, Markov processes and their functionals in duality, Z. Wahrscheinlichkeitstheorie und verw. Gebiete 24 (1972), 229-246. MR 48:7398

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 60J65, 60J55

Retrieve articles in all journals with MSC (1991): 60J65, 60J55

Additional Information

P. J. Fitzsimmons
Affiliation: Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093–0112

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

American Mathematical Society