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UCSD MATHEMATICS DEPARTMENT: PROBABILITY SEMINAR
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**
Jason Swanson (Univ. of Washington) **

The p-th Variation of a Brownian Martingale with an Application to Mathematical Finance
It is well known that a continuous martingale M(t)
has a finite quadratic
variation, which is independent of time partitions used. Moreover, the
p-th variation of M(t) is zero if p>2 and infinity if p<2.
For a continuous martingale M(t) that is adapted to a
Brownian filtration and for p other than 2, suitably rescaling
the p-th variation of M(t) will result in nontrivial limits.
Unlike the p=2 case, however, the limit depends on the choice of
the time partitions. I will discuss what the rescaling is, what
the limit is, and how it depends on the time partitions.
The special case p=1 will be used to partially generalize a
result of Grannan and Swindle regarding the scaled limit of
transaction costs in a model of mathematical finance.