# short talks

Saturday, November 4, 2006 - 10:40am - 11:40am

**Asymptotics of eigenvalue clusters for Schroedinger operators on**

the Sierpinski gasket

Kasso Okoudjou (University of Maryland)

In this talk we shall present some results on the asymptotic

behavior of spectra of Schrodinger operators with continuous potential on

the Sierpinski gasket SG. In particular, using the extence of localized

eigenfunctions for the Laplacian on SG we show that the eigenvalues of

the Schrodinger opeartor break into clusters around certain eigenvalue of

the Laplacian. Moreover, we prove that the characteristic measure of

these clusters converges to a measure.**Option pricing with memory**

Flavia Sancier-Barbosa (Southern Illinois University)

In this talk we introduce an option pricing model with

delayed memory. The memory is introduced in the stock

dynamics, which is described by a stochastic

functional differential equation. The model has the

following key features:

1. Volatility depends on a (delayed) history, i.e.,

its value at time t is a deterministic functional of

the history of the stock from time t-L up to time t-l,

where l is positive and less than or equal to L.

Hence, due to this past-dependence on the stock price,

the volatility is necessarily stochastic.

2. The randomness in the volatility is intrinsic,

since it is generated by past values of the stock

price.

3. The stock dynamics is driven by a single

one-dimensional Brownian motion, and the model is one

dimensional.

4. The market is complete.

5. For large delays (or at times relatively close to

maturity) we obtain a closed-form representation for

the fair price of the option, as well as for the

hedging strategy.

6. The option price can be expressed in terms of the

exact solution of a one-dimensional partial

differential equation (PDE).

7. The classical Black-scholes model is a particular

case of the delayed memory model.

8. We believe that our model is sufficiently flexible

to fit real market data, in particular to account for

observed smiles and frowns.**Probabilistic and stochastic modeling of turbulent flows**

Sean Garrick (University of Minnesota, Twin Cities)

The transport of wide variety of phenomena in turbulent flows (heat,

mass, momentum, species, etc.) is a significant challenge to

computational scientists and engineers working in chemical

processing, pharmaceuticals, materials synthesis, and atmospheric

physics, to name a few. Capturing the variety of length and time

scales manifest in these flows leads to compute times which are

impractical at best and infeasible at worst. In this seminar, I will

present some ideas and recent work in the modeling of multi-scale

transport phenomena and the probabilistic and stochastic tools used

in their description.