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
3. The stock dynamics is driven by a single
one-dimensional Brownian motion, and the model is one
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
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.