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.