Past

We study an equilibrium model for a defaultable bond in the asymmetric dynamic information setting. The market consists of noise traders, an insider and a risk neutral market maker. Under the assumption that the insider observes the firm value continuously in time we study the optimal strategies for the insider and the optimal pricing rules for the market maker. We show that there exists an equilibrium where the insider’s trades are inconspicuous. In this equilibrium the insider drives the total demand to a certain level at the default time. The solution follows from answering the following purely mathematical question which is of interest in its own: Suppose Z and B are two independent Brownian motions with B(0)=0 and Z(0) is a positive random variable. Let T be the first time that Z hits 0. Does there exists a semimartingale X such that

1) it is a solution to the SDE

dX(t) = dB(t) + g(t,X(t),Z(t))dt

with X(0) = 1, for some appropriate function g,

2) T is the first hitting time of 0 for X, and

3) X is a Brownian motion in its own filtration?

• Alfio Quarteroni (EPFL) — Heterogeneous Domain Decomposition Methods
• Laure Saint-Raymond (Paris VI & ENS) — Weak compactness methods for singular penalization problems with boundary layers
• Bryce McLeod (Oxford) — A problem in dislocation theory
• Tom Bridges (Surrey) — Degenerate conservation laws, bifurcation of solitary waves and the concept of criticality in fluid mechanics
• Neshan Wickramasekera (Cambridge) — Frequency functions and singular set bounds for branched minimal graphs

The meeting is being held in the Mary Ogilvie Lecture Theatre, St Anne’s College and will start promptly at 9:30am with the last talk finishing at 4:30pm.

For the full programme and registration pages please see: http://www2.maths.ox.ac.uk/oxpde/meetings/

The Schur product is the commutative operation of entrywise

multiplication of two (possibly infinite) matrices. If we fix a matrix

A and require that the Schur product of A with the matrix of any

bounded operator is again the matrix of a bounded operator, then A is

said to be a Schur multiplier; Schur multiplication by A then turns

out to be a completely bounded map. The Schur multipliers were

characterised by Grothendieck in the 1950s. In a 2006 paper, Kissin

and Shulman study a noncommutative generalisation which they call

"operator multipliers", in which the theory of operator spaces plays

an important role. We will present joint work with Katja Juschenko,

Ivan Todorov and Ludmilla Turowska in which we determine the operator

multipliers which are completely compact (that is, they satisfy a

strengthening of the usual notion of compactness which is appropriate

for completely bounded maps).

This is the second of two seminars this afternoon describing a generalization of Donaldson-Thomas invariants, joint work of Yinan Song and Dominic Joyce. (Still work in progress.)

Behrend showed that conventional Donaldson-Thomas invariants can be written as the Euler characteristic of the moduli space of semistable sheaves weighted by a "microlocal obstruction function" \mu.

In previous work, the speaker defined Donaldson-Thomas type invariants "counting" coherent sheaves on a Calabi-Yau 3-fold using

Euler characteristics of sheaf moduli spaces, and more generally, of moduli spaces of "configurations" of sheaves. However, these invariants are not deformation-invariant.

We now combine these ideas, and insert Behrend's microlocal obstruction \mu into the speaker's previous definition to get new generalized Donaldson-Thomas invariants. Microlocal functions \mu have a multiplicative property implying that the new invariants transform according to the same multiplicative transformation law as the previous invariants under change of stability condition.

Then we show that the invariants counting pairs in the previous seminar are sums of products of the new generalized Donaldson-Thomas invariants. Since the pair invariants are deformation invariant, we can deduce by induction on rank that the new generalized Donaldson-Thomas invariants are unchanged under deformations of the underlying Calabi-Yau 3-fold.

For any simple graph G and any positive integer lambda, let

P(G,lambda) denote the number of mappings f from V(G) to

{1,2,..,lambda} such that f(u) not= f(v) for every two adjacent

vertices u and v in G. It can be shown that

P(G,lambda) = \sum_{A \subseteq E} (-1)^{|A|} lambda^{c(A)}

where E is the edge set of G and c(A) is the number of components

of the spanning subgraph of G with edge set A. Hence P(G,lambda)

is really a polynomial of lambda. Many results on the chromatic

polynomial of a graph have been discovered since it was introduced

by Birkhoff in 1912. However, there are still many unsolved

problems and this talk will introduce the progress of some

problems and also some new problems proposed recently.

This is the first of two seminars this afternoon describing a generalization of Donaldson-Thomas invariants, joint work of Yinan Song and Dominic Joyce. We shall define invariants "counting" semistable coherent sheaves on a Calabi-Yau 3-fold. Our invariants are invariant under deformations of the complex structure of the underlying Calabi-Yau 3-fold, and have known transformation law under change of stability condition.

This first seminar constructs an auxiliary invariant "counting" stable pairs (s,E), where E is a Gieseker semistable coherent sheaf with fixed Hilbert polynomial and s : O(-n) --> E for n >> 0 is a morphism of sheaves, and (s,E) satisfies a stability condition. Using Behrend-Fantechi's approach to obstruction theories and virtual classes we prove this auxiliary invariant is unchanged under deformation of the underlying Calabi-Yau 3-fold.

I will begin by talking briefly about the Lavrentiev phenomenon and its implications for computations. In short, if a minimization problem exhibits a Lavrentiev gap then `naive' numerical methods cannot be used to solve it. In the past, several regularization techniques have been used to overcome this difficulty. I will briefly mention them and discuss their strengths and weaknesses.

The main part of the talk will be concerned with a class of convex problems, and I will show that for this class, relatively simple numerical methods, namely (i) the Crouzeix--Raviart FEM and (ii) the P2-FEM with under-integration, can successfully overcome the Lavrentiev gap.

I will start by reviewing the current status of the stability

problem for black holes in general relativity. In the second part of the

talk I will focus on a particular (symmetry) class of five-dimensional

dynamical black holes recently introduced by Bizon et al as a model to

study gravitational collapse in vacuum. In this context I state a recent

result establishing the asymptotic stability of the five dimensional

Schwarzschild metric with respect to vacuum perturbations in the given

class.

We construct a kinetic SDE in the state variables (position,velocity), where the spatial dependency in the drift term of the velocity equation is a conditional expectation with respect to the position. Those systems are introduced in fluid mechanic by S. B. Pope and are used in the simulation of complex turbulent flows. Such simulation approach is known as Probability Density Function (PDF) method .

We construct a PDF method applied to a dynamical downscaling problem to generate fine scale wind : we consider a bounded domain D. A weather prediction model solves the wind field at the boundary of D (coarse resolution). In D, we adapt a Lagrangian model to the atmospheric flow description and we construct a particles algorithm to solve it (fine resolution).

In the second part of the talk, we give a (partial) construction of a Lagrangian SDE confined in a given domain and such that the corresponding Eulerian velocity at the boundary is given. This problem is related to stochastic impact problem and existence of trace at the boundary for the McKean-Vlasov equations with specular boundary condition

The Cameron-Martin theorem is a fundamental result in stochastic analysis. We will show that the Wiener measure on a geometrically and stochastically complete Riemannian manifold is quasi-invariant. This is a complete a complete generalization of the classical Cameron-Martin theorem for Euclidean space to Riemannian manifolds. We do not impose any curvature growth conditions.

We consider a financial contract that delivers a single cash flow given by the terminal value of a cumulative gains process.

The problem of modelling such an asset and associated derivatives is important, for example, in the determination of optimal insurance claims reserve policies, and in the pricing of reinsurance contracts. In the insurance setting, aggregate claims play the role of cumulative gains, and the terminal cash flow represents the totality of the claims payable for the given accounting period. A similar example arises when we consider the accumulation of losses in a credit portfolio, and value a contract that pays an amount equal to the totality of the losses over a given time interval. An expression for the value process of such an asset is derived as follows. We fix a probability space, together with a pricing measure, and model the terminal cash flow by a random variable; next, we model the cumulative gains process by the product of the terminal cash flow and an independent gamma bridge; finally, we take the filtration to be that generated by the cumulative gains process.

An explicit expression for the value process is obtained by taking the discounted expectation of the future cash flow, conditional on the relevant market information. The price of an Arrow–Debreu security on the cumulative gains process is determined, and is used to obtain a closed-form expression for the price of a European-style option on the value of the asset at the given intermediate time. The results obtained make use of remarkable properties of the gamma bridge process, and are applicable to a wide variety of financial products based on cumulative gains processes such as aggregate claims, credit portfolio losses, defined benefit pension schemes, emissions, and rainfall. (Co-authors: D. C. Brody, Imperial College London, and A.

Macrina, King's College London and ETH Zurich. Downloadable at

www.mth.kcl.ac.uk.