國家理論科學研究中心學術演講
NCTS Seminar on Probability

Speaker:          (1) Mr. Guan-Yu Chen

                             (Ph.D. student , Cornell University & NCTU)

                        (2) Professor Ching-Tang Wu 吳慶堂

                             (National University of Kaohsiung)

Topic     :         (1) Eichler-Shimura Isomorphism                              

                        (2) Periods of Cusp Form

Time      :         June 10 (Fri.)  (I) 9:00~10:30   (II) 11:00~12:00

Abstracts:

The standard riffle shuffle of Gilbert, Shannon and Reeds (GSR-shuffle) is a model for the way good card players shuffle cards. This model was introduced by Gilbert and Shannon in 1955 and independently by Reeds in 1981. Roughly, a GSR-shuffle is first cutting a deck of $n$ cards into two piles according to an $(n,\frac{1}{2})$ -binomial random variable and then dropping cards one by one from one or the other pack with probability proportional to the relative sizes of the packs. Aldous shows, in 1983, that asymptotically as $n$, the number of cards, tends to infinity, it takes $\frac{3}{2}\log_2n$ shuffles to mix up the deck if convergence is measured in total variation. Bayer and Diaconis obtained in 1992 an exact useful formula for the probability distribution of deck arrangements after $k$ GSR-shuffle and derived the "shape" and "window" of the cutoff for GSR-shuffle.

Here we consider some generalizations of the standard GSR-shuffle. This is a joint work with Laurant Saloff-Coste and we generalize GSR-shuffle as follows: Instead of cutting a deck into two piles binomially, we first select a positive integer $m$ according to a probability measure and then shuffle cards with an $m$-shuffle. In this talk, we show that the total variation cutoff is still preserved and the "window" of the cutoff is optimized under some specific assumptions. An interesting observation to be introduced is that under some circumstances, the total variation cutoff is presented in such a sharp way that it happens only in one or two steps. (II) Robust utility functionals arise as numerical representations of investor preferences, when the investor is uncertain about the underlying probabilistic model and averse against both risk and model uncertainty. In this talk, we study the duality theory for the problem of maximizing the robust utility of the terminal wealth in a general incomplete merket model.