The Annals of Probability

Brownian Motion with Lower Class Moving Boundaries Which Grow Faster Than $t^{1/2}$

R. F. Bass and M. Cranston

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Abstract

Upper and lower bounds are obtained for $P(|W(t)| \leq f(t), t \leq u)$ and $P(|S(n)| \leq f(n), n \leq N), u, N$ large, where $W(t)$ is a Brownian motion, $S(n)$ is a random walk with $ES(1) = 0, E|S(1)|^{2+2\eta} < \infty$, and $f(t)$ is a deterministic function growing faster than $t^{1/2}$ but slower than $(2t \ln \ln t)^{1/2}$.

Article information

Source
Ann. Probab., Volume 11, Number 1 (1983), 34-39.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993657

Digital Object Identifier
doi:10.1214/aop/1176993657

Mathematical Reviews number (MathSciNet)
MR682798

Zentralblatt MATH identifier
0503.60080

JSTOR
links.jstor.org

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60J15 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Keywords
Moving boundaries Brownian motion random walks

Citation

Bass, R. F.; Cranston, M. Brownian Motion with Lower Class Moving Boundaries Which Grow Faster Than $t^{1/2}$. Ann. Probab. 11 (1983), no. 1, 34--39. doi:10.1214/aop/1176993657. https://projecteuclid.org/euclid.aop/1176993657


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