Mon, 10 May 2004
14:15
DH 3rd floor SR

Small time behaviour of double stochastic integrals and hedging under gamma constraints

Touzi Nizar
Abstract

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.

Fri, 07 May 2004
14:15
DH 3rd floor SR

TBA

Christoph Reisinger
(Oxford)
Thu, 06 May 2004

14:00 - 15:00
Comlab

Nonhydrodynamic modes and lattice Boltzmann equations with general equations of state

Dr Paul Dellar
(University of Oxford)
Abstract

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.

Mon, 03 May 2004
15:45
DH 3rd floor SR

The Brownian snake and random trees

Svante Janson
(University of Uppsala)
Abstract

The Brownian snake (with lifetime given by a normalized

Brownian excursion) arises as a natural limit when studying random trees. This

may be used in both directions, i.e. to obtain asymptotic results for random

trees in terms of the Brownian snake, or, conversely, to deduce properties of

the Brownian snake from asymptotic properties of random trees. The arguments

are based on Aldous' theory of the continuum random tree.

I will discuss two such situations:

1. The Wiener index of random trees converges, after

suitable scaling, to the integral (=mean position) of the head of the Brownian

snake. This enables us to calculate the moments of this integral.

2. A branching random walk on a random tree converges, after

suitable scaling, to the Brownian snake, provided the distribution of the

increments does not have too large tails. For i.i.d increments Y with mean 0,

a necessary and sufficient condition is that the tails are o(y^{-4}); in

particular, a finite fourth moment is enough, but weaker moment conditions are

not.

Mon, 03 May 2004
14:15
DH 3rd floor SR

An extension of Levy-Khinchine formula in semi-Dirichlet forms setting

Ma Zhi-Ming
Abstract

The celebrated Levy-Khintchine formula provides us an explicit

structure of Levy processes on Rd. In this talk I shall present a

structure result for quasi-regular semi-Dirichlet forms, i.e., for

those semi-Dirichlet forms which are associated with right processes

on general state spaces. The result is regarded as an extension of

Levy-Khintchine formula in semi-Dirichlet forms setting. It can also

be regarded as an extension of Beurling-Deny formula which is up to

now available only for symmetric Dirichlet forms.