# East Midlands Stochastic Analysis Seminar sponsored by the London Mathematical Society and Department of Mathematics, University of York

Series:
Mathematical Finance and Stochastic Analysis Seminar
Date and time:
July 30, 2012, 14:00 - 18:00
Speaker:
David Applebaum (Sheffield), David Kelly (Warwick), Daniel Ueltschi (Warwick)
Room:
G/020

Abstracts:

D. Ueltschi: The spin correlations in quantum Heisenberg systems can be represented by certain random loop models, as was shown by Toth (ferromagnet) and Aizenman-Nachtergaele (antiferromagnet). Namely, spin correlations are given by the probability that two lattice sites belong to the same loop. In dimensions 3 and more, one expects a transition to a phase with macroscopic loops; this is supported by a famous result of Dyson-Lieb-Simon. I will arguethat there are many large loops, and that their joint distribution is given by Poisson-Dirichlet. I will present numerical evidence in some related situations.

D. Applebaum: We obtain martingale transforms that are built in a natural way from a generic Levy process in a  Lie group $G$. Using sharp inequalities due to Burkholder we then construct a class of linear operators that are bounded on $L^{p}(G,m)$ (where $m$ is a Haar measure) for all $1 < p < \infty$. When the group is compact, we use Peter-Weyl theory to exhibit specific examples of these operators  as Fourier multipliers. These include second order Riesz transforms, imaginary powers  of the Laplacian and operators associated with subordinated Brownian motion.  Talk based on joint work with Rodrigo Banuelos (Purdue). The  paper is available at  http://arxiv.org/abs/1206.1560

D. Kelly: The theory of rough paths provides a powerful framework in which stochastic differential equations can be solved deterministically. Until recently, this theory could only deal with Stratonovich-type integrals or in the new parlance, equations driven by geometric rough paths. In order to treat equations involving Ito-type integrals (or indeed any not necessarily Stratonovich-type integrals) one needs a theory of non-geometric rough paths. This was provided recently by Gubinelli, who introduced the idea of branched rough paths.
In this talk we will review the basic idea of branched rough paths and how they can be used to solve SDEs. We will then provide a kind of conversion theorem between branched rough paths and geometric rough paths. In particular, we show that every branched rough path can be written as a geometric rough path lying above an extension of the original driving path. As an easy corollary, we obtain a generalised Ito-Stratonovich conversion formula, whereby every equation involving non-geometric integrals can be rewritten as an extended equation involving only geometric integrals.

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