Forthcoming Seminars › Seminar Series

Stochastic Analysis Seminar

Mon, 14/01/2008
13:15 		Prof. Cedric Villani (ENS Lyon) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
Optimal transport and curvature (monge meets Riemann)
Born in France around 1780, the optimal transport problem has known a scientific explosion in the past two decades, in relation with dynamical systems and partial differential equations. Recently it has found unexpected applications in Riemannian geometry, in particular the encoding of Ricci curvature bounds
Mon, 14/01/2008
14:45 		Prof. Olivier Raimond (Universite Paris-Sud XI) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
On some generalized reinforced random walks on integers
This is a joint work with Bruno Schapira, and it is a work in progress. We study recurrence and transience properties of some edge reinforced random walks on the integers: the probability to go from $ x $ to $ x+1 $ at time $ n $ is equal to $ f(\alpha_n^x) $ where $ \alpha_n^x=\frac{1+\sum_{k=1}^n 1_{(X_{k-1},X_k)=(x,x+1)}}{2+\sum_{k=1}^n 1_{X_{k-1}=x}} $. Depending on the shape of $ f $, we give some sufficient criteria for recurrence or transience of these walks
Mon, 21/01/2008
01:15 		Prof. Istvan Gyongy (Edinburgh) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
Accelerated finite difference schemes
Some recent joint results with N. V. Krylov on the convergence of solutions of finite difference schemes are presented. The finite difference schemes, considered in the talk correspond to discretizations (in the space variable) of second order parabolic and of second order elliptic (possibly degenerate) equations. Space derivatives of the solutions to the finite difference schemes are estimated, and these estimates are applied to show that the convergence of finite difference approximations for equations in the whole space can be accelerated to any given rate. This result can be applied to stochastic PDEs, in particular to the Zakai equation of nonlinear filtering, when the signal and observation noises are independent.
Mon, 21/01/2008
13:15 		Prof. Istvan Gyongy (Edinburgh) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
Accelerated finite difference schemes
Some recent joint results with N. V. Krylov on the convergence of solutions of finite difference schemes are presented. The finite difference schemes, considered in the talk correspond to discretizations (in the space variable) of second order parabolic and of second order elliptic (possibly degenerate) equations. Space derivatives of the solutions to the finite difference schemes are estimated, and these estimates are applied to show that the convergence of finite difference approximations for equations in the whole space can be accelerated to any given rate. This result can be applied to stochastic PDEs, in particular to the Zakai equation of nonlinear filtering, when the signal and observation noises are independent.
Mon, 21/01/2008
14:45 		Prof. Franck Barthe (Universite Paul Sabatier) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
Isoperimetric bounds under curvature and integrability assumptions
The Bakry Emery criterion asserts that a probability measure with a strictly positive generalized curvature satisfies a logarithmic Sobolev inequality, and by results of Bakry and Ledoux an isoperimetric inequality of Gaussian type. These results were complemented by a theorem of Wang: if the curvature is bounded from below by a negative number, then under an additional Gaussian integrability assumption, the log-Sobolev inequality is still valid. The goal of this joint work with A. Kolesnikov is to provide an extension of Wang's theorem to other integrability assumptions. Our results also encompass a theorem of Bobkov on log-concave measures on normed spaces and allows us to deal with non-convex potentials when the convexity defect is balanced by integrability conditions. The arguments rely on optimal transportation and its connection to the entropy functional
Mon, 28/01/2008
13:15 		Prof. Philippe Bougerol (Universite Pierre et Marie Curie) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
Brownian paths and Representation theory
Counting paths, or walks, is an important ingredient in the classical representation theory of compact groups. Using Brownian paths gives a new flexible and intuitive approach, which allows to extend some of this theory to the non- cristallographic case. This is joint work with P. Biane and N. O'Connell
Mon, 28/01/2008
14:45 		Prof. Jiangang Ying (Fudan University) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
The Feynman-Kac formula and related problems
This talk gives a survey on a series of work which I and co-authors have been doing for 10 years. I will start from the Feynman-Kac type formula for Dirichlet forms. Then a necessary and sufficient condition is given to characterize the killing transform of Markov processes. Lastly we shall discuss the regular subspaces of linear transform and answer some problems related to the Feynman-Kac formula
Mon, 04/02/2008
13:15 		Prof. Bernt Oksendal (Universitetet i Oslo) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
A Malliavin calculus approach to a general maximum principle for stochastic control of jump diffusions
The classical maximum principle for optimal control of solutions of stochastic differential equations (developed by Pontryagin (deterministic case), Bismut, Bensoussan, Haussmann and others), assumes that the system is Markovian and that the controller has access to full, updated information about the system at all times. The classical solution method involves an adjoint process defined as the solution of a backward stochastic differential equation, which is often difficult to solve. We apply Malliavin calculus for Lévy processes to obtain a generalized maximum principle valid for non-Markovian systems and with (possibly) only partial information available for the controller. The backward stochastic differential equation is replaced by expressions involving the Malliavin derivatives of the quantities of the system. The results are illustrated by some applications to finance
Mon, 04/02/2008
14:45 		Dr David Steinsaltz (Oxford) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
Asymptotics of killed Markov processes, with applications to the biodemography of ageing
The convergence of Markov processes to stationary distributions is a basic topic of introductory courses in stochastic processes, and the theory has been thoroughly developed. What happens when we add killing to the process? The process as such will not converge in distribution, but the survivors may; that is, the distribution of the process, conditioned on survival up to time t, converges to a "quasistationary distribution" as t goes to infinity. This talk presents recent work with Steve Evans, proving an analogue of the transience-recurrence dichotomy for killed one-dimensional diffusions. Under fairly general conditions, a killed one-dimensional diffusion conditioned to have survived up to time t either escapes to infinity almost surely (meaning that the probability of finding it in any bounded set goes to 0) or it converges to the quasistationary distribution, whose density is given by the top eigenfunction of the adjoint generator. These theorems arose in solving part of a longstanding problem in biological theories of ageing, and then turned out to play a key role in a very different problem in population biology, the effect of unequal damage inheritance on population growth rates.
Mon, 11/02/2008
13:15 		Dr Harry Zheng (London) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
Constrained Non-smooth Utility Maximization without Quadratic Inf-convolution
In this talk we revisit the setting of Bouchard, Touzi, and Zeghal (2004). For an incomplete market and a non-smooth utility function U defined on the whole real line we study the problem: sup E [U(XTx,θ – B)] θΘ(S) Here B is a bounded contingent claim and Xx,θ represents the wealth process with initial capital x generated by portfolio θ. We study the case when the portfolios are constrained in a closed convex cone. For the case without constraints and with a smooth utility function the solution method is to approximate the utility function and look at the same problem on a bounded negative domain. However, when one attempts to solve this bounded domain problem for a non-smooth utility function, the standard methods of proof cannot be applied. To circumvent this difficulty the idea of quadratic inf-convolution was introduced in Bouchard, Touzi, and Zeghal (2004). This method is mathematically appealing but leads to lengthy and technical proofs. We will show that despite the presence of constraints, the dependence on quadratic inf-convolution can be removed. We will also show the existence of a constrained replicating portfolio for the optimal terminal wealth when the filtration is generated by a Brownian motion. This provides a natural generalisation of the results of Karatzas and Shreve (1998) to the whole real line.
Mon, 11/02/2008
14:45 		Prof. Yuri Kondratiev (University of Reading) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
Stochastic competition models from ecology to society
We describe individual based continuous models of random evolutions and discuss some effects of competitions in these models. The range of applications includes models of spatial ecology, genetic mutation-selection models and particular socio-economic systems. The main aim of our presentation is to establish links between local characteristics of considered models and their macroscopic behaviour
Mon, 18/02/2008
13:15 		Prof. Peter Bank (Technische Universitat Berlin) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
A large investor trading at market indifference prices
Mon, 18/02/2008
14:45 		Dr Jonathan Jordan (Sheffield) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
A preferential duplication random graph and links to vertex reinforced random walks
Mon, 25/02/2008
13:15 		Dr Silke Rolles (Munchen, Germany) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
Linearly edge-reinforced random walks, part I
We consider a linearly edge-reinforced random walk on a class of two-dimensional graphs with constant initial weights. The graphs are obtained from Z^2 by replacing every edge by a sufficiently large, but fixed number of edges in series. We prove that a linearly edge-reinforced random walk on these graphs is recurrent. Furthermore, we derive bounds for the probability that the edge-reinforced random walk hits the boundary of a large box before returning to its starting point. Part I will also include an overview on the history of the model. In part II, some more details about the proofs will be explained.
Mon, 25/02/2008
14:45 		Dr Franz Merkl (Munchen, Germany) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
Linearly edge-reinforced random walks, part II
We consider a linearly edge-reinforced random walk on a class of two-dimensional graphs with constant initial weights. The graphs are obtained from Z^2 by replacing every edge by a sufficiently large, but fixed number of edges in series. We prove that a linearly edge-reinforced random walk on these graphs is recurrent. Furthermore, we derive bounds for the probability that the edge-reinforced random walk hits the boundary of a large box before returning to its starting point. Part I will also include an overview on the history of the model. In part II, some more details about the proofs will be explained.
Mon, 03/03/2008
13:15 		Dr Christina Goldschmidt (Department of Statistics, Oxford) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
The allele frequency spectrum associated with the Bolthausen-Sznitman coalescent
I will take as my starting point a problem which is classical in population genetics: we wish to understand the distribution of numbers of individuals in a population who carry different alleles of a certain gene. We imagine a sample of size n from a population in which individuals are subject to neutral mutation at a certain constant rate. Every mutation gives rise to a completely new type. The genealogy of the sample is modelled by a coalescent process and we imagine the mutations as a Poisson process of marks along the coalescent tree. The allelic partition is obtained by tracing back to the most recent mutation for each individual and grouping together individuals whose most recent mutations are the same. The number of blocks of each of the different possible sizes in this partition is called the allele frequency spectrum. Recently, there has been much interest in this problem when the underlying coalescent process is a so-called Lambda-coalescent (even when this is not a biologically “reasonable” model) because the allelic partition is a nice example of an exchangeable random partition. In this talk, I will describe the asymptotics (as n tends to infinity) of the allele frequency spectrum when the coalescent process is a particular Lambda-coalescent which was introduced by Bolthausen and Sznitman. It turns out that the frequency spectrum scales in a rather unusual way, and that we need somewhat unusual tools in order to tackle it. This is joint work with Anne-Laure Basdevant (Toulouse III).
Mon, 03/03/2008
14:45 		Prof. Balint Toth (Budapest) Stochastic Analysis Seminar Add to calendar Oxford-Man Institute
Some new results on 1-d self-repelling random walks
I will present two new results in the context of the title. Both are joint work with B. Veto. 1. In earlier work a limit theorem with $ t^{2/3} $ scaling was established for a class of self repelling random walks on $ \mathbb Z $ with long memory, where the self-interaction was defined in terms of the local time spent on unoriented edges. For combinatorial reasons this proof was not extendable to the natural case when the self-repellence is defined in trems of local time on sites. Now we prove a similar result for a *continuous time* random walk on $ \mathbb Z $, with self-repellence defined in terms of local time on sites. 2. Defining the self-repelling mechanism in terms of the local time on *oriented edges* results in totally different asymptotic behaviour than the unoriented cases. We prove limit theorems for this random walk with long memory.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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