Past

We prove that when a sequence of Lévy processes $X(n)$ or a normed sequence of random walks $S(n)$ converges a.s. on the Skorokhod space toward a Lévy process $X$, the sequence $L(n)$ of local times at the supremum of $X(n)$ converges uniformly on compact sets in probability toward the local time at the supremum of $X$. A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and

descending) converges jointly in law towards the ladder processes of $X$. As an application, we show that in general, the sequence $S(n)$ conditioned to stay positive converges weakly, jointly with its local time at the future minimum, towards the corresponding functional for the limiting process $X$. From this we deduce an invariance principle for the meander which extends known results for the case of attraction to a stable law.

Defaults in a credit portfolio of many obligors or in an economy populated with firms tend to occur in waves. This may simply reflect their sharing of common risk factors and/or manifest their systemic linkages via credit chains. One popular approach to characterizing defaults in a large pool of obligors is the Poisson intensity model coupled with stochastic covariates, or the Cox process for short. A constraining feature of such models is that defaults of different obligors are independent events after conditioning on the covariates, which makes them ill-suited for modeling clustered defaults. Although individual default intensities under such models can be high and correlated via the stochastic covariates, joint default rates will always be zero, because the joint default probabilities are in the order of the length of time squared or higher. In this paper, we develop a hierarchical intensity model with three layers of shocks -- common, group-specific and individual. When a common (or group-specific) shock occurs, all obligors (or group members) face individual default probabilities, determining whether they actually default. The joint default rates under this hierarchical structure can be high, and thus the model better captures clustered defaults. This hierarchical intensity model can be estimated using the maximum likelihood principle. A default signature plot is invented to complement the typical power curve analysis in default prediction. We implement the new model on the US corporate bankruptcy data and find it far superior to the standard intensity model both in terms of the likelihood ratio test and default signature plot.

We consider the three dimensional Navier-Stokes equations with a large initial data and

we prove the existence of a global smooth solution. The main feature of the initial data

is that it varies slowly in the vertical direction and has a norm which blows up as the

small parameter goes to zero. In the language of geometrical optics, this type of

initial data can be seen as the ``ill prepared" case. Using analytical-type estimates

and the special structure of the nonlinear term of the equation we obtain the existence

of a global smooth solution generated by this large initial data. This talk is based on a

work in collaboration with J.-Y. Chemin and I. Gallagher and on a joint work with Z.

Zhang.

The aim is to explore whether we can extend the work of PM Woodward first published many years ago, to see if we can extract more information than we do to date from our radar returns. A particular interest is in the information available for target recognition, which requires going beyond Woodward's assumption that the target has no internal structure.

CFD is an indispensible part of the design process for all major gas turbine components. The growth in the use of CFD from single-block structured mesh steady state solvers to highly resolved unstructured mesh unsteady solvers will be described, with examples of the design improvements that have been achieved. The European Commission has set stringent targets for the reduction of noise, emissions and fuel consumption to be achieved by 2020. The application of CFD to produce innovative designs to meet these targets will be described. The future direction of CFD towards whole engine simulations will also be discussed.

I will survey the theory of quasiHamiltonian spaces, a.k.a. group valued moment maps. In rough correspondence with historical development, I will first show how they emerge from the study of loop group representations, and then how they arise as a special case of "presymplectic realizations" in Dirac geometry.

HTML clipboard /*-->*/ /*-->*/ We introduce a "Polya choice" urn model combining elements of the well known "power of two choices" model and the "rich get richer" model. From a set of $k$ urns, randomly choose $c$ distinct urns with probability proportional to the product of a power $\gamma>0$ of their occupancies, and increment one with the smallest occupancy. The model has an interesting phase transition. If $\gamma \leq 1$, the urn occupancies are asymptotically equal with probability 1. For $\gamma>1$, this still occurs with positive probability, but there is also positive probability that some urns get only finitely many balls while others get infinitely many.