Brownian space-time functions of zero quadratic variation depend only on time
HTML articles powered by AMS MathViewer
- by P. J. Fitzsimmons PDF
- Proc. Amer. Math. Soc. 127 (1999), 2423-2429 Request permission
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
- Patrick J. Fitzsimmons, Even and odd continuous additive functionals, Dirichlet forms and stochastic processes (Beijing, 1993) de Gruyter, Berlin, 1995, pp. 139–154. MR 1366430
- P. J. Fitzsimmons, Drift transformations of symmetric diffusions and duality, Preprint, 1997.
- Hans Föllmer, Philip Protter, and Albert N. Shiryayev, Quadratic covariation and an extension of Itô’s formula, Bernoulli 1 (1995), no. 1-2, 149–169. MR 1354459, DOI 10.2307/3318684
- Kiyosi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Springer-Verlag, Berlin-New York, 1974. Second printing, corrected. MR 0345224
- F. B. Knight, On a Brownian motion problem of T. Salisbury, Canad. Math. Bull. 40 (1997), 67–71.
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1994. MR 1303781
- Thomas S. Salisbury, An increasing diffusion, Seminar on stochastic processes, 1984 (Evanston, Ill., 1984) Progr. Probab. Statist., vol. 9, Birkhäuser Boston, Boston, MA, 1986, pp. 173–194. MR 896729, DOI 10.1007/978-1-4684-6745-1_{1}1
- Michael Sharpe, General theory of Markov processes, Pure and Applied Mathematics, vol. 133, Academic Press, Inc., Boston, MA, 1988. MR 958914
- Hiroshi Tanaka, Note on continuous additive functionals of the $1$-dimensional Brownian path, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1 (1962/63), 251–257. MR 169307, DOI 10.1007/BF00532497
- John B. Walsh, Markov processes and their functionals in duality, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24 (1972), 229–246. MR 329056, DOI 10.1007/BF00532535
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
- 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1487367