# Prob-L@B

(Probability Laboratory at Bath)

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Alex Cox
Email: A.M.G.Cox@bath.ac.uk

### Prob-L@b Seminars: Spring 2013

Our seminars our usually held at 12.15 p.m. on Mondays in room 4W 1.7 . If you wish to find out more, please contact one of the organisers. The speakers are internal (Bath) unless otherwise stated. Details of previous semesters can be found here

4/2/13: Martin Forde (King's College, London)

Model-independent bounds for general path dependent options - a convex duality approach

CANCELLED - NEW DATE TBC

11/2/13: Tiziano De Angelis (Manchester)

Optimal stopping of a Hilbert space valued diffusion and applications to finance

Pricing American Bond options in a market model with forward interest rates corre- sponds, in mathematical terms, to an optimal stopping problem of an infinite dimensional diffusion. Motivated by this financial application we analyse a finite horizon optimal stopping problem for an infinite dimensional diffusion $$X$$ by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space $$H$$ with a non-linear diffusion coefficient $$\sigma(X)$$ and a generic unbounded operator $$A$$ in the drift term. We show that when the gain function $$\Theta$$ is time-dependent and fulfils mild regularity assumptions, the value function $$U$$ of the optimal stopping problem solves an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. The solution of the variational problem is found in a suitable Banach space $$V$$ fully characterized in terms of a Gaussian measure $$\mu$$ associated to the coefficient $$\sigma(X)$$.

18/2/13: Stefan Grosskinsky (Warwick)

Dynamics of condensation in inclusion processes

The inclusion process is an interacting particle system where particles on connected sites attract each other in addition to performing independent random walks. The system has stationary product measures and exhibits condensation in the limit of strong interactions, where all particles concentrate on a single lattice site. We study the equilibration dynamics on finite lattices in the limit of infinitely many particles, which, in addition to jumps of whole clusters, contains an interesting continuous mass exchange between clusters given by Wright-Fisher diffusions. During equilibration the number of clusters decreases monotonically, and the stationary dynamics consist of jumps of a single remaining cluster (the condensate). This is joint work with Frank Redig and Kiamars Vafayi.

25/2/13: Dafydd Evans (Cardiff)

Fast anomaly detection in spatial point processes

CANCELLED - NEW DATE TBC

4/3/12: Johannes Ruf (Oxford)

Nonnegative local martingales, Novikov's and Kazamaki's criteria, and the distribution of explosion times

I will give a new proof for the famous criteria by Novikov and Kazamaki, which provide sufficient conditions for the martingale property of a nonnegative local martingale. The proof is based on an extension theorem for probability measures that can be considered as a generalization of a Girsanov-type change of measure. In the second part of my talk I will illustrate how a generalized Girsanov formula can be used to compute the distribution of the explosion time of a weak solution to a stochastic differential equation.
Parts of this talk are based on joined working papers with Martin Larsson and Ioannis Karatzas.

11/3/13: Antal Jarai

Electrical resistance of the low-dimensional critical branching random walk

We consider the trace of a critical branching random walk in $$d+1$$ dimensions conditioned to survive forever. We show that the electrical resistance between the origin and generation $$n$$ grows sublinearly in $$n$$ when $$d<6$$. In particular, it follows that in $$d=5$$ the spectral dimension of simple random walk on the trace is strictly larger than 4/3, answering a question of Barlow, Jarai, Kumagai and Slade. (Joint work with Asaf Nachmias.)

20/3/13: Charles Bordenave (Toulouse)

Spectrum of Markov generators on sparse random graphs

In this talk, we will consider various probability distributions on the set of stochastic matrices with n states and on the set of Laplacian/Kirchhoff matrices on n states. They will arise naturally from the conductance model on n states with i.i.d conductances. With the help of random matrix theory, we will study the spectrum of these processes.

NOTE: Unusual time and date. Wednesday, 14.15 in 1E2.4

8/4/13: Tim Rogers

Demographic noise leads to the spontaneous formation of species

In this talk I will discuss an evolutionary model of competition, which is a microscopic stochastic analogue of a famous population-level model in ecology. I will show how the effects of demographic noise in the stochastic model give rise to radically different macro-scale behaviour. The (non-rigorous) analysis uses an expansion in system size, coupled with a time-scale separation argument.

15/4/13: David Applebaum (Sheffield)

Brownian motion, martingale transforms and Fourier multipliers on Lie groups.

We associate a space-time martingale to Brownian motion on a Lie group $$G$$ and transform it to obtain a family of "differentially subordinate" martingales. Using powerful inequalities dues to Burkholder, Banuelos and Wang we construct a family of linear operators which are bounded on $$L^{p}(G, \tau)$$ (where $$\tau$$ is a Haar measure) for all $$1 < p < \infty$$. When $$G$$ is compact, we can utilise non-commutative Fourier analysis to represent these operators as Fourier multipliers. Examples include second order Riesz transforms and operators of Laplace transform type.

Talk based on joint work with Rodrigo Banuelos

22/4/13: Dafydd Evans (Cardiff)

Fast anomaly detection in spatial point processes

29/4/13: Yan Fyodorov (Queen Mary, London)

Fluctuations and extreme values in multifractal patterns

The goal is to understand sample-to-sample fluctuations in disorder-generated multifractal intensity patterns. Arguably the simplest model of that sort is the exponential of an ideal periodic 1/f Gaussian noise. The latter process can be looked at as a one-dimensional "projection" of 2D Gaussian Free Field and inherits from it the logarithmic covariance structure. It most naturally emerges in the random matrix theory context, but attracted also an independent interest in statistical mechanics of disordered systems. I will determine the threshold of extreme values of 1/f noise and provide a rather compelling explanation for the mechanism behind its universality. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields.
The presentation will be mainly based on the joint work with Pierre Le Doussal and Alberto Rosso, J Stat Phys: 149 (2012), 898-920 as well as on some related earlier works by the speaker.

13/5/13: Denis Denisov (Manchester)

Tail behaviour of stationary distribution for Markov chains with asymptotically zero drift

We consider a one-dimensional Markov chain with asymptotically zero drift and finite second moments of jumps which is positive recurrent. A power-like asymptotic behaviour of the invariant tail distribution is proven; such a heavy-tailed invariant measure happens even if the jumps of the chain are bounded. Our analysis is based on test functions technique and on construction of a harmonic function. This is a joint work with Korshunov and Wachtel.

20/5/13: Peter Mörters

Emergence of condensation in models of selection and mutation

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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