Room S27 in the Statistical Laboratory
The canonical time is 2.05pm on Tuesdays.
All interested are encouraged
to take part to the full by presenting their ideas and discussing
those of others. Graduate students especially are urged to attend.
Tuesday January 19
2.05pm Gesine Reinert
An Introduction to Stein's Method, and Applications. I
In the first session we will introduce Stein's method, as developed by
Charles Stein in 1972. As this method has been proven particularly
powerful for normal approximations, for Poisson and Compound Poisson
approximations, and for laws of large numbers, we will concentrate on
these types of limit results. We will see that Stein's method is
particularly useful in deriving bounds on the rates of convergence. As
this session will introduce the main ideas of Stein's method, we will
restrict ourselves to real-valued random elements.
Tuesday January 26
2.05pm Gesine Reinert
An Introduction to Stein's Method, and Applications. II
In the second session we will develop a framework for measure-valued
random elements, together with metrics that describe different types of
convergence. In this framework we will derive Stein's method for the law
of large numbers, for Gaussian approximations for measure-valued random
elements and for Poisson point process approximation. Examples from
combinatorics, from computational biology and from stochastic processes
are given.
Tuesday February 2
2.05pm Yuval Peres (Jerusalem)
Percolation on groups: progress and problems
Tuesday February 9
2.05pm Ben Hambly (BRIMS)
Spectral asymptotics for random recursive fractals
Tuesday February 16
2.05pm Geoffrey Grimmett
Weak convergence and probabilistic number theory
`Modern' methods of probability theory may be applied
to study classical problems of number theory, such as the
distributional properties of number-theoretic sieves.
One may obtain weak convergence and large deviation theorems,
but some central questions remain open. Stein's method may find
applications here.
FRIDAY February 19
2.05pm James Taylor (Sussex)
The multifractal structure of Galton-Watson branching measure
Branching measure ($\mu$) can be thought of as the result
of a flow down the tree in which a unit
mass at the root divides among the branches
each time it comes to a fork. This produces a random
measure on the space of infinite paths from the
root of the tree. A natural metric on the space of paths
is given by $d(x.y) = exp (-n)$, where $n$ is the
largest level such that $x$ and $y$ have common
paths from the root to level $n$. If $m > 1$,
it is known that, on a typical path $x$ of the ball $B(x,r)$ of
radius $r$ centred on $x$ is of order $r^\alpha$
as $r$ decreases to $0$. The multifractal problem
seeks to investigate the exceptional sets
where $(\mu)B(x,r)$ behaves like $r^\beta$ as $r\rightarrow
0$. For $\beta$ different to $\alpha$ such
sets will have zero measure, but we should decide when
they are not empty - and then how big they are in some appropriate sense.
Tuesday February 23
2.05pm James Norris
Small time fluctuations of the Brownian bridge
A diffusion process conditioned to pass from $x$ to $y$ in time $t$
must, for small $t$, stay close to a geodesic from $x$ to $y$
for the Riemannian metric associated with the generator.
In this talk I will discuss how stochastic differential equations
allow one to study the fluctuations of the diffusion about the
geodesic. There is a Gaussian limit which may be expressed in
terms of a standard Brownian bridge and the curvature along the geodesic.
Tuesday March 2
2.05pm Peter Grandits
On Leland's approach to option pricing under transaction costs
A very interesting approach in the literature to the
problem of hedging (forming a selffinancing portfolio, which replicates the payo
ff
of an option at exercise date)
under transaction costs is by Leland
(1985). He claims that
even in the presence of transaction costs a call option on a stock $S$,
described by geometric Brownian motion, can be perfectly hedged using
Black-Scholes delta hedging with a modified volatility. Recently
Kabanov and Safarian (1997)
disproved this claim, giving an explicit (up to an double integral) expression
of the limiting (the numbers of revision intervals tending to infinity)
hedging error. It appears to be strictly negative
and depends on the path of the
stock
price only via the stock price at expiry $S_T$. We will show
that the limiting hedging error, considered as a function of $S_T$, exhibits
a removable discontinuity at the exercise price $K$. Furthermore, we provide a
quantitative result describing the evolution of the discontinuity,
which shows that its precursors can very well be
observed also in cases of reasonable length
of revision intervals.
Tuesday March 9
2.05pm Yoav Git
Martingales vs large deviations for a
2-D branching diffusion model
(Joint work with Simon Harris, sch@maths.bath.ac.uk, Bath)
The framework we work on is a (position-dependent) branching diffusion
process where our aim is to find the spread of particles in space.
Finding a family of martingales associated with the BDP usually does
the trick, (we can think of them as the change-of-measure which
gives the large-deviations lower bound).
However, we discuss a 2-D branching diffusion process first
introduced by Simon Harris and David Williams where the
right martingales only give partial lower-bounds.
We discover that to be a typical particle
(in any given position), a particle must have had an
atypical history.
Room S27 in the Statistical Laboratory
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