Past

We calculate an analytic value for the correlation coefficient between a geometric, or exponential, Brownian motion and its time-average, a novelty being our use of divided differences to elucidate formulae. This provides a simple approximation for the value of certain Asian options regarding them as exchange options. We also illustrate that the higher moments of the time-average can be expressed neatly as divided differences of the exponential function via the Hermite-Genocchi integral relation, as well as demonstrating that these expressions agree with those obtained by Oshanin and Yor when the drift term vanishes.

Galois groups of p-class towers of number fields have long been a mystery,

but recent calculations have led to glimpses of a rich theory behind them,

involving Galois actions on trees, families of groups whose derived series

have finite index, families of deficiency zero p-groups approximated by

p-adic analytic groups, and so on.

We consider semilinear Sturm-Liouville and elliptic problems with jumping

nonlinearities. We show how `half-eigenvalues' can be used to describe the

solvability of such problems and consider the structure of the set of

half-eigenvalues. It will be seen that for Sturm-Liouville problems the

structure of this set can be considerably more complicated for periodic than

for separated boundary conditions, while for elliptic partial differential

operators only partial results are known about the structure in general.

We construct spaces of manifolds of various dimensions following

Vassiliev's approach to the theory of knots. These are infinite-dimensional

spaces with hypersurface, corresponding to manifolds with Morse singularities.

Connected components of the complement to this discriminant are homotopy

equivalent to the covering spaces of BDiff(M). These spaces appear to be a

natural base over which one can consider parametrised versions of Floer and

Seiberg-Witten theories.

The formation of steady patterns in one space dimension is generically

governed, at small amplitude, by the Ginzburg-Landau equation.

But in systems with a conserved quantity, there is a large-scale neutral

mode that must be included in the asymptotic analysis for pattern

formation near onset. The usual Ginzburg-Landau equation for the amplitude

of the pattern is then coupled to an equation for the large-scale mode.

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These amplitude equations show that for certain parameters all regular

periodic patterns are unstable. Beyond the stability boundary, there

exist stable stationary solutions in the form of spatially modulated

patterns or localised patterns. Many more exotic localised states are

found for patterns in two dimensions.

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Applications of the theory include convection in a magnetic field,

providing an understanding of localised states seen in numerical

simulations.

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.

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.

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.

The celebrated Levy-Khintchine formula provides us an explicit

structure of Levy processes on $R^d$. 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.

Consider approximating a set of discretely defined values $f_{1}, \ldots , f_{m}$ say at $x=x_{1}, x_{2}, \ldots, x_{m}$, with a chosen approximating form. Given prior knowledge that noise is present and that some might be outliers, a standard least squares approach based on $l_{2}$ norm of the error $\epsilon$ may well provide poor estimates. We instead consider a least squares approach based on a modified measure of the form $\tilde{\epsilon} = \epsilon (1+c^{2}\epsilon^{2})^{-\frac{1}{2}}$, where $c$ is a constant to be fixed.

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The choice of the constant $c$ in this estimator has a significant effect on the performance of the estimator both in terms of its algorithmic convergence to a solution and its ability to cope effectively with outliers. Given a prior estimate of the likely standard deviation of the noise in the data, we wish to determine a value of $c$ such that the estimator behaves like a robust estimator when outliers are present but like a least squares estimator otherwise.

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We describe approaches to determining suitable values of $c$ and illustrate their effectiveness on approximation with polynomial and radial basis functions. We also describe algorithms for computing the estimates based on an iteratively weighted linear least squares scheme.

In Clarendon Lab