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WROCŁAW UNIVERSITY
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TECHNOLOGY

Contents of PMS, Vol. 21, Fasc. 1,
pages 199 - 212
 

ASYMPTOTICS OF THE SUPREMUM OF SCALED BROWNIAN MOTION

Krzysztof Dębicki

Abstract: We consider the problem of estimating the tail of the distribution of the supremum of scaled Brownian motion B(f (t)) processes with linear drift.

Using the local time technique we obtain asymptotics and bounds of

P (sup(B(f (t))- t) > u), t>t0
which are expressed in terms of the expected value of the local time of B(f(t))- t processes at level u.

As an application we obtain upper bounds for the tail of distribution of the supremum for some Gaussian processes with stationary increments.

1991 AMS Mathematics Subject Classification: Primary 60G15, Secondary 60G70, 68M20.

Key words and phrases: Brownian motion, exponential bound, fractional Brownian motion, Gaussian process, local time, scaled Brownian motion.

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