Department of Mathematical Sciences

Dini's helix - a pseudospherical surface Brownian motion Willmore cylinder with umbilic lines
		   (Babich-Bobenko) Triadic Von Koch Snowflake - Fleckinger, Levitin
		   and Vassiliev Darboux transform of a Clifford torus (Holly
		   Bernstein) Mandelbrot fractal geometry

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Alex Cox

Informal Probability Seminars: Spring 2009

Our seminars our usually held at 12.15 p.m. on Wednesdays in room 1W 3.6 . If you wish to find out more, please contact one of the organisers. The speakers are mostly internal (Bath) unless otherwise stated. Details of previous semesters can be found here

28/1/09: Pavel Gapeev (LSE)

Perpetual American options in models with default risk and random dividends

We present closed form solutions to the problems of pricing of perpetual American standard options in two diffusion models of financial markets with presence of default risk. The method of proof is based on reducing the initial discounted optimal stopping problems to equivalent free-boundary problems and solving the latter by means of the smooth-fit conditions. Applying the recently derived change-of-variable formula with local time on surfaces, we verify that the obtained solutions of the free-boundary problems turn out to be solutions of the initial optimal stopping problems.
In the first model, it is assumed that a firm can spontaneously announce a default and change the value of the dividend rate at that time. Being inaccessible for usual investors trading at the market, such a change in the dividend rate can be actually hidden into the behaviour of the firm value under the risk-neutral measure. Aiming at explicit expressions for the rational option prices and related exercise boundaries, we assume exponential distribution for the default time at which the dividend rate changes from one constant value to another, within the framework of a diffusion model for the firm value dynamics. We embed the initial problem into a discounted optimal stopping problem for a two-dimensional Markov process, having the firm value and conditional survival probability as components. It turns out that the latter process is equivalent to the posterior probability of the occurrence of 'disorder' in the corresponding problem of detecting a change in the drift rate of an observed Wiener process. By means of solving the equivalent free-boundary problem, we show that the exercise boundary for the firm value is constant and can be characterized as a unique solution of a transcendental equation.
In the second model, it is assumed that the default happens when the firm value falls to some random barrier, which is not observable from the market. Aiming at closed form expressions for ex-dividend prices and exercise boundaries, we assume that the barrier has exponential distribution and it is independent of the firm value, which is modelled by a geometric Brownian motion. We embed the initial problem into an optimal stopping problem for a two-dimensional Markov process having the firm value and its running minimum as components. Note that the resulting problem turns out to be necessarily two-dimensional in the sense that it cannot be reduced to a problem for a one-dimensional Markov process. We show that the optimal exercise boundary for the firm value can be expressed as a function of the running minimum process. By means of solving the equivalent free-boundary problem, where the normal reflection condition at the diagonal of the two-dimensional state space breaks down, we show that the exercise boundary can be characterized as a unique solution of a nonlinear ordinary differential equation.

4/2/09: Parkpoom Phetpradap

Large deviations for the range of a simple random walk

12/2/09: Wolfgang König (Leipzig)

A Variational Formula for the Free Energy of a Many-Boson System

Note time and day: 11.15am, Thursday

18/2/09: Mladen Savov (Paris VI)

Right Inverses for Levy Processes

An integral criterion in terms of the Levy triplet is proposed which tests whether a Levy Process has a right inverse, i.e. there exists a subordinator K(x) such that X(K(x))=x at least for each x in (0,a), where a>0 is a random variable. This completes previous studies by Evans and Winkel.

25/2/09: Vitali Wachtel (Munich)

Heavy-traffic analysis of the maximum of an asymptotically stable random walk

For families of random walks $\{S_k^{(a)}\}$ with $\mathbf E S_k^{(a)} = -ka < 0$ we consider their maxima $M^{(a)} = \sup_{k \ge 0} S_k^{(a)}$. We investigate the asymptotic behaviour of $M^{(a)}$ as $a \to 0$ for asymptotically stable random walks. This problem appeared first in the 1960's in the analysis of a single-server queue when the traffic load tends to $1$ and since then is referred to as the heavy-traffic approximation problem. Kingman and Prokhorov suggested two different approaches which were later followed by many authors. We give two elementary proofs of our main result, using each of these approaches. It turns out that the main technical difficulties in both proofs are rather similar and may be resolved via a generalisation of the Kolmogorov inequality to the case of an infinite variance.

4/3/09: Stanislav Volkov (Bristol)

Going through a passport control with wife, or sequential adsorption at extremes.

In a simple version of a cooperative sequential adsorption model, particles consecutively arrive on the set of vertices {1,2,...,M} uniformly spaced on a circumference. A particle arriving at time t=0,1,2,... gets attached to a vertex j with probability proportional to beta^{N(t,j)} where N(t,j) is a number of particles already attached to the vertices in a certain neighbourhood of vertex j. Examples of such a neighbourhood are:
(1) the vertex itself and its left neighbour (asymmetric case);
(2) the vertex itself and its left and right neighbours (symmetric case).
We are interested in the long-time behaviour of this process. Our most recent results cover the cases when beta=0 and beta=+infinity, the situations which can be quite naturally interpreted in a queueing theory setup
The talk is a sequel to the seminar given by Vadim Shchervakov on the 29 October 2008, and will highlight the most recent findings based on our joint work.

11/3/09: Mathew Penrose

Normal approximation for an allocation model

Consider throwing n balls independently into m boxes of possibly non-equal size (probability). Let S be the resulting number of isolated balls. We discuss error bounds in the normal approximation for S, and bounds for the variance of S in terms of the underlying parameters. Advanced prerequisites are at a minimum for this talk.

18/3/09: Matthias Winkel (Oxford)

A new family of Markov branching trees: the alpha-gamma model

We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this paper, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour.

Note time and location: 12.45, 6E 2.1

8/4/09: Günter Last (Karlsruhe)

Compound Poisson asymptotics for level crossings of piecewise deterministic Markov processes

We consider a piecewise-deterministic Markov process governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. First we discuss a version of Rice's formula relating the stationary density of the process to level crossing intensities. Under additional assumptions we will then show that a suitably time-scaled point process of level crossings can be approximated by a geometrically compound Poisson process.
The talk is based on joint work with Kostya Borovkov.

29/4/09: Manfred Salmhofer (Heidelberg)

Diffusion of a quantum particle in a random environment

6/5/09: Pierre Tarrès (Oxford)

Brownian polymers

We consider a process $X_t\in\R^d$, $t\ge0$, introduced by Durrett and Rogers in 1992 in order to model the shape of a growing polymer; it undergoes a drift which depends on its past trajectory, and a Brownian increment. Our work concerns two conjectures by these authors (1992), concerning repulsive interaction functions $f$ in dimension $1$ ($\forall x\in\R$, $xf(x)\ge0$).
We showed the first one with T. Mountford (AIHP, 2008), for certain functions $f$ with heavy tails, leading to transience to $+\infty$ or $-\infty$ with probability $1/2$. We partially proved the second one with B. Tóth (preprint), for rapidly decreasing functions $f$, through a study of the local time environment viewed from the particule: we explicitly display an associated invariant measure, which enables us to prove under certain initial conditions that $X_t/t\to_{t\to\infty}0$, and that the process is at least diffusive asymptotically.

13/5/09: Frank Aurzada (TU Berlin)

Small deviations of general Lévy processes

The small deviation problem for a stochastic process $(X_t)_{t\in[0,1]}$ consists in determining the rate of the quantity $$ P( \sup_{t\in[0,1]} |X_t| \leq \varepsilon ) $$ when $\varepsilon\to 0$. This problem has several connections to approximation quantities and the path regularity of the processes. The talk starts with a short overview of the field of small deviation probabilities and its applications.
Then recent results are presented when $X$ is a real-valued Lévy processes. In this case, it is possible to obtain tight estimates for the above probability (on the exponential scale) only in terms of the characterizing triplet $(c,\sigma2,\nu)$ of $X$. Several examples are discussed as well as ideas of the proofs, which include exponential change of measure and exit time estimates. In the multi-dimensional setting, the problem is open, and I briefly describe the arising difficulties.
This is joint work with Steffen Dereich (Berlin).

20/5/09: Tom Rosoman

Critical probabilities in continuum percolation

27/5/09: Marcel Ortgiese

Ageing in the parabolic Anderson model


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