Probability Seminar, Statistical Laboratory

Probability Seminar, Statistical Laboratory

Tuesdays 4.30-5.30 pm, MR 12.
Organisers: Nathanaël Berestycki and Mike Tehranchi.

Webpage of the probability group

Tea beforehand at 4pm in the common room, and drinks afterwards at the Punter.

Easter 2010

Previous term

Date Speaker Title Notes
27 April Nicolas Curien (ENS) Recursive triangulations and fragmentation theory (abstract)
4 May Nathanael Berestycki (Cambridge) Asymptotic behaviour of near-critical branching Brownian motion (abstract)
5 May Stanislav Smirnov (University of Geneva) Discrete complex analysis and probability (details) Rollo Davidson lecture. 5pm in Wolfson lecture room in CMS.
11 May Simon Harris (Bath) Branching Brownian motion in an inhomogeneous breeding potential (abstract). Cancelled due to illness
11 May Jan Kallsen (University of Kiel) On a Heath-Jarrow-Morton approach for stock options (abstract).
18 may Jakob Bjornberg (Cambridge) Stochastic geometry of the quantum Ising model (abstract). First talk of the day, at 3pm.
18 May Thaleia Zariphopoulou (Oxford) (abstract). Second talk of the day, at 4.30pm
25 May Arne Lokka (LSE) Detection of critical events before public announcements (abstract).
15 June Eric Shellef (Weizmann Institute) On the range of a random walk in a torus (abstract).

Abstract:

27 April, Nicolas Curien. Recursive triangulations and fragmentation theory. Abstract:

Consider the convex regular polygon with n vertices. A triangulation of this polygon is a set of n-3 non crossing diagonals that completely triangulates it. We study random triangulations obtained by adding diagonals progressively, and in particular not sampled from the uniform measure. We establish the convergence of these random triangulations towards a random closed subset of Hausdorff dimension (\sqrt{17}-3)/2.

4 May, Nathanael Berestycki. Asymptotic behaviour of near-critical branching Brownian motion. Abstract:

Consider a system of particles that perform branching Brownian motion with negative drift \sqrt(2- \eps) and are killed upon hitting zero. Initially, there is just one particle at x. Kesten (1978) proved that the system survives if and only if \eps>0. In this talk I shall describe recent joint work with Julien Berestycki and Jason Schweinsberg concerning the limiting behaviour of this process as \eps tends to 0. In particular we establish sharp asymptotics for the limiting survival probability as a function of the starting point x. Moreover, the limiting genealogy between individuals from this population is shown to have a characteristic time scale of order \eps^{-3/2}. When time is measured in these units we show that the geometry of the genealogical tree converges to the Bolthausen-Sznitman coalescent. This is closely related to a set of conjectures by Brunet, Derrida and Simon.

11 May Jan Kallsen. Abstract:

In the Heath-Jarrow-Morton (HJM) approach in interest rate theory the whole forward rate curve rather than the short rate is considered as state variable for a stochastic model. Absence of arbitrage then leads to consistency and drift restrictions, in particular the HJM drift condition. Several attempts have been made to transfer this idea to options on a stock, cf. e.g. by Schönbucher (1999), Schweizer & Wissel (2008), Carmona & Nadtochiy (2009), Jacod & Protter (2006). Here, the underlying stock plays the role of the short rate. The implied volatility surface or a reparametrisation serves as state variable and hence as counterpart of the forward rate curve in the classical framework of HJM. Our approach to this problem resembles Carmona & Nadtochiy (2009) in that we try to preserve main features of the HJM setup. However, it is based on a different parametrisation or codebook, which allows to simplify both theory and application.

18 June, Jakob Bjornberg. Stochastic geometry of the quantum Ising model. Abstract:

The classical Ising model is a well-studied model for phase transition. A particularly fruitful way to study this model is via its `graphical representations', which allow us to use ideas and arguments from stochastic geometry and percolation theory. This talk will focus on graphical representations of the QUANTUM Ising model, a much less studied topic. In joint work with Geoffrey Grimmett, we were able to use graphical representations to deduce fundamental facts about the quantum Ising model. Most notable of these are sharpness of the phase transition in all dimensions, and bounds on some critical exponents.

25 May, Arne Lokka. Detection of critical events before public announcements. Abstract:

I consider an asset price which follows a geometric Brownian motion, but which changes its drift an some unobservable time before a random observable time (which could correspond to the announcement of a takeover/merger). This change in behaviour has been documented in the literature and can be attributed to insider trading. I derive the dynamics under incomplete information and use a change point detection formulation to find the optimal time to sell/buy the stock.

15 June, Eric Shellef. On the Range of a Random Walk In a Torus. Abstract:

Let a random walk run inside a torus of dimension three or higher for a number of steps which is a constant proportion of the volume. We examine geometric properties of the range, the random subgraph induced by the set of vertices visited by the walk. We prove local distance and mixing bounds for the typical range that are a k-iterated log factor from those on the full torus for arbitrary k. The proof uses hierarchical renormalization and techniques that can be applied to more general random processes in the Euclidean lattice.