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X-WR-CALNAME:Ph.D. Oral Comprehensive Examination
X-WR-TIMEZONE:America/New_York
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X-LIC-LOCATION:America/New_York
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DTSTART:19700308T020000
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DTSTART:19701101T020000
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20141007T154000
DTEND;TZID=America/New_York:20141007T174000
LOCATION:MB114
URL:http://www.math.psu.edu/seminars/meeting.php?id=25352
SUMMARY:Ph.D. Oral Comprehensive Examination - "Local time of discrete stoc
hastic processes and a homogenization problem”
DESCRIPTION:Seminar: Ph.D. Oral Comprehensive Examination\nTitle: "Local ti
me of discrete stochastic processes and a homogenization problem”\nSpeak
er: Xiaofei Zheng\, Adviser: Manfred Denker\, Penn State\nAbstract: Suppo
se {X_i} are i.i.d or strictly stationary\, S_n is the partial sum of {X_i
}. We are interested in studying the time that S_n spends at a certain lev
el and its limiting behavior. For continuous stochastic processes\, it is
described by local time. Levy first introduced Brownian local time in 1939
. For discrete processes\, we are looking for a good definition of local t
ime. We defined the local time of a random walk\, it turns out that after
a proper scaling\, the local time converges to the Brownian local time. Fo
r discrete martingale\, I will introduce how to embed it into the path of
a Brownian motion\, and study the downcrossing number. In the second part
of my talk\, I will introduce the problem of homogenization driven by frac
tional Brownian motion. Based on the work of Iyer\, Komorowski\, Novikov a
nd Ryzhik\, we conjecture that the solution to dX_t=-AV(X_t)dt+dB^H(t) con
verges to Brownian motion weakly. The possible method is to introduce stop
ping times and study the number of downcrossings.
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