Past Forthcoming Seminars

10 May 2004
Touzi Nizar
We formulate a problem of super-hedging under gamma constraint by taking the portfolio process as a controlled state variable. This leads to a non-standard stochastic control problem. An intuitive guess of the associated Bellman equation leads to a non-parabolic PDE! A careful analysis of this problem leads to the study of the small time behaviour of double stochastic integrals. The main result is a characterization of the value function of the super-replication problem as the unique viscosity solution of the associated Bellman equation, which turns out to be the parabolic envelope of the above intuitive guess, i.e. its smallest parabolic majorant. When the underlying stock price has constant volatility, we obtain an explicit solution by face-lifting the pay-off of the option.
  • Stochastic Analysis Seminar
6 May 2004
Dr Paul Dellar
The lattice Boltzmann equation has been used successfully used to simulate nearly incompressible flows using an isothermal equation of state, but much less work has been done to determine stable implementations for other equations of state. The commonly used nine velocity lattice Boltzmann equation supports three non-hydrodynamic or "ghost'' modes in addition to the macroscopic density, momentum, and stress modes. The equilibrium value of one non-hydrodynamic mode is not constrained by the continuum equations at Navier-Stokes order in the Chapman-Enskog expansion. Instead, we show that it must be chosen to eliminate a high wavenumber instability. For general barotropic equations of state the resulting stable equilibria do not coincide with a truncated expansion in Hermite polynomials, and need not be positive or even sign-definite as one would expect from arguments based on entropy extremisation. An alternative approach tries to suppress the instability by enhancing the damping the non-hydrodynamic modes using a collision operator with multiple relaxation times instead of the common single relaxation time BGK collision operator. However, the resulting scheme fails to converge to the correct incompressible limit if the non-hydrodynamic relaxation times are fixed in lattice units. Instead we show that they must scale with the Mach number in the same way as the stress relaxation time.
  • Computational Mathematics and Applications Seminar