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: Winter 2007

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

10/10/07: Zaeem Burq (Melbourne)

One-dimensional Brownian motion crossing non-smooth curves

17/10/07: Victor Rivero

On the asymptotic behaviour of increasing self-similar Markov processes.

It has been proved by Bertoin and Caballero in 2002 that an increasing self-similar Markov process (issMp) has polynomial growth, in the sense of convergence in distribution, if and only if the increasing Levy process associated to it via Lamperti's transformation has finite mean. The purpose of this talk is to explain the asymptotic behaviour of an issMp in the case where the underlying increasing Levy process has infinite mean. To that end I will start by providing a short introduction to the theory of positive self-similar Markov processes (pssMp), then I will explain the transformation introduced by Lamperti relating the class of Levy processes with that of pssMp. To finish, I will present some recent results of the type central limit theorem and law of iterated logarithm about the rate of growth at infinity of the logarithm of an increasing self-similar Markov processes.

24/10/07: Steffen Dereich

Random networks with concave preferentia attachment rule: Degree evolutions

31/10/07: Peter Mörters

A tale of two cities: the parabolic Anderson model with heavy-tailed potential

7/11/07: Tobias Mueller (EURANDOM/Technische Universiteit Eindhoven)

Two-point concentration in random geometric graphs

21/11/07: Alex Cox

Optimal Skorokhod Embeddings: extremal range probabilities

The Skorokhod embedding problem is to find a stopping time of a Brownian motion such that the stopped process has a specified distribution. Many solutions are known, including the solution of Azéma and Yor, which additionally maximises the probability of the maximium being greater than a given level among all Skorokhod embeddings of a given distribution. Motivated by applications to finance, we construct embeddings which maximise and minimise the probability of both passing above a given level, and passing below a given level. (Joint work with Jan Obłój, Imperial College.)

5/12/07: Ronnie Loeffen

On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Levy processes

The classical optimal dividends control problem introduced by de Finetti has been extensively studied in the Cramer-Lundberg (C-L) risk model. Recently Avram, Palmowski & Pistorius considered the case were the risk process is modelled by a general spectrally negative Levy process. Up till now, an explicit solution to this control problem has only been found in two concrete examples: the C-L model with exponentially distributed claims and the case where the risk process is modelled by a Brownian motion with drift. In both cases the optimal strategy is formed by a barrier strategy. In this talk, we show that if an easy-to-check analytical condition on the Levy measure is satisfied, then the optimal strategy is always of the barrier type. In the analysis the so-called scale functions of spectrally negative Levy processes play a central role.

12/12/07: Robert Knobloch

Uniform conditional ergodicity and intrinsic ultracontractivity

19/12/07: Renming Song(Illinois)

Boundary Harnack principle for subordinate Brownian motions

In this talk I will present some resent progress in the potential theory of subordinate Brownian motions. In particular I will show that the boundary Harnack principle holds for a large class of subordinate Brownian motions.
This class of subordinate Brownian motion can not be regarded as perturbations of symmetric stable processes.


Seminars will be added to this list as they are confirmed. Please check back for the latest list, or subscribe to the prob-sem mailing list to receive details of future seminars

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