Abstract
We revisit a result of Uchiyama (1980):giv en that a certain integral test is satisfied, the rate of the probability that Brownian motion remains below the moving boundary f is asymptotically the same as for the constant boundary. The integral test for f is also necessary in some sense.
After Uchiyama’s result, a number of different proofs appeared simplifying the original arguments, which strongly rely on some known identities for Brownian motion. In particular, Novikov (1996) gives an elementary proof in the case of an increasing boundary. Here, we provide an elementary, halfpage proof for the case of a decreasing boundary. Further, we identify that the integral test is related to a repulsion effect of the three-dimensional Bessel process. Our proof gives some hope to be generalized to other processes such as FBM.
Mathematics Subject Classification (2010). Primary 60G15; Secondary 60G18.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
M.D. Bramson, Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31(5) (1978) 531–581.
J. Gärtner, Location of wave fronts for the multidimensional KPP equation and Brownian first exit densities. Math. Nachr. 105 (1982) 317–351.
C. Jennen and H.R. Lerche, First exit densities of Brownian motion through onesided moving boundaries. Z. Wahrsch. Verw. Gebiete 55(2) (1981) 133–148.
I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics Springer-Verlag, New York, second edition, 1991.
A.A. Novikov, A martingale approach to first passage problems and a new condition for Wald’s identity. Stochastic differential systems (Visegrád, 1980), volume 36 of Lecture Notes in Control and Information Sci., pages 146–156. Springer, Berlin, 1981.
A.A. Novikov, Martingales, a Tauberian theorem, and strategies for games of chance. Teor. Veroyatnost. i Primenen. 41(4) (1996) 810–826.
D. Slepian, The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 (1962) 463–501.
K. Uchiyama, Brownian first exit from and sojourn over one-sided moving boundary and application. Z. Wahrsch. Verw. Gebiete 54(1) (1980) 75–116.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this paper
Cite this paper
Aurzada, F., Kramm, T. (2013). First Exit of Brownian Motion from a One-sided Moving Boundary. In: Houdré, C., Mason, D., Rosiński, J., Wellner, J. (eds) High Dimensional Probability VI. Progress in Probability, vol 66. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0490-5_13
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0490-5_13
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0489-9
Online ISBN: 978-3-0348-0490-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)