Oxford-Man Institute

Let u be a vector field on a bounded domain in R^3. The absolute boundary condition states that both the normal part of u and the tangential part of curl(u) vanish on the boundary. After motivating the use of this condition in the context of the Navier Stokes equation, we prove local (in time) existence with this boundary behaviour. This work is together with Dr. Z. Qian and Prof. G. Q. Chen, Northwestern University.

We will analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. These results will be applied to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdos-Renyi graphs and graphs with prescribed degree distribution. We will sketch examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices. If time allows, we will discuss related open problems. This is a joint work with Marc Lelarge (INRIA & Ecole Normale Supérieure).

Many portfolio optimization problems are directly or indirectly concerned with the current maximum of the underlying. For example, loockback or Russian options, optimization with max-drawdown constraint , or indirectly American Put Options, optimization with floor constraints.

The Azema-Yor martingales or max-martingales, introduced in 1979 to solve the Skohorod embedding problem, appear to be remarkably efficient to provide simple solution to some of these problems, written on semi-martingale with continuous running supremum.

A trader in finance is called an insider if she (or he) knows more about the prices in the market than can be obtained from the market history itself. This is the case if, for example, the trader knows something about the future price/value of a stock. We discuss the following question: What is the optimal portfolio of an insider who wants to maximize her expected profit at a given future time? The problem is that heavy trading by the insider will reveal parts of her inside price information to the market and thereby reduce her information advantage.

We will solve this problem by presenting a general anticipative stochastic calculus model for insider trading. Our results generalize equilibrium results due to Kyle (1985) and Back (1992).

The presentation is partly based on recent joint work with Knut Aase and Terje Bjuland, both at the Norwegian School of Economics and Business Administration (NHH).

Some recent joint results with N. V. Krylov on the convergence of solutions of finite difference schemes are presented.

The finite difference schemes, considered in the talk correspond to discretizations (in the space variable) of second order parabolic and of second order elliptic (possibly degenerate) equations.

Space derivatives of the solutions to the finite difference schemes are estimated, and these estimates are applied to show that the convergence of finite difference approximations for equations in the whole space can be accelerated to any given rate. This result can be applied to stochastic PDEs, in particular to the Zakai equation of nonlinear filtering, when the signal and observation noises are independent.

I will present two new results in the context of the title. Both are joint work with B. Veto.

1. In earlier work a limit theorem with $t^{2/3}$ scaling was established for a class of self repelling random walks on $\mathbb Z$ with long memory, where the self-interaction was defined in terms of the local time spent on unoriented edges. For combinatorial reasons this proof was not extendable to the natural case when the self-repellence is defined in trems of local time on sites. Now we prove a similar result for a *continuous time* random walk on $\mathbb Z$, with self-repellence defined in terms of local time on sites.

2. Defining the self-repelling mechanism in terms of the local time on *oriented edges* results in totally different asymptotic behaviour than the unoriented cases. We prove limit theorems for this random walk with long memory.

I will take as my starting point a problem which is classical in

population genetics: we wish to understand the distribution of numbers

of individuals in a population who carry different alleles of a

certain gene. We imagine a sample of size n from a population in

which individuals are subject to neutral mutation at a certain

constant rate. Every mutation gives rise to a completely new type.

The genealogy of the sample is modelled by a coalescent process and we

imagine the mutations as a Poisson process of marks along the

coalescent tree. The allelic partition is obtained by tracing back to

the most recent mutation for each individual and grouping together

individuals whose most recent mutations are the same. The number of

blocks of each of the different possible sizes in this partition is

called the allele frequency spectrum. Recently, there has been much

interest in this problem when the underlying coalescent process is a

so-called Lambda-coalescent (even when this is not a biologically

``reasonable'' model) because the allelic partition is a nice example

of an exchangeable random partition. In this talk, I will describe

the asymptotics (as n tends to infinity) of the allele frequency

spectrum when the coalescent process is a particular Lambda-coalescent

which was introduced by Bolthausen and Sznitman. It turns out that

the frequency spectrum scales in a rather unusual way, and that we

need somewhat unusual tools in order to tackle it.

This is joint work with Anne-Laure Basdevant (Toulouse III).

We consider a linearly edge-reinforced random walk

on a class of two-dimensional graphs with constant

initial weights. The graphs are obtained

from Z^2 by replacing every edge by a sufficiently large, but fixed

number of edges in series.

We prove that a linearly edge-reinforced random walk on these graphs

is recurrent. Furthermore, we derive bounds for the probability that

the edge-reinforced random walk hits the boundary of a large box

before returning to its starting point.

Part I will also include an overview on the history of the model.

In part II, some more details about the proofs will be explained.

We consider a linearly edge-reinforced random walk

on a class of two-dimensional graphs with constant

initial weights. The graphs are obtained

from Z^2 by replacing every edge by a sufficiently large, but fixed

number of edges in series.

We prove that a linearly edge-reinforced random walk on these graphs

is recurrent. Furthermore, we derive bounds for the probability that

the edge-reinforced random walk hits the boundary of a large box

before returning to its starting point.

Part I will also include an overview on the history of the model.

In part II, some more details about the proofs will be explained.