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

    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
    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.