Eagle House

We analyze the fluctuation of the loss from default around its large portfolio limit in a class of reduced-form models of correlated default timing. We prove a weak convergence result for the fluctuation process and use it for developing a conditionally Gaussian approximation to the loss distribution. Numerical results illustrate the accuracy of the approximation.

This is joint work with Kostas Spiliopoulos (Boston University) and Justin Sirignano (Stanford).

Azema martingales arise naturally in the study of the chaotic representation property; they also provide classical interpretations of quantum stochastic calculus. The talk will not insist on these aspects, but only define these processes and address the problem of their classification. This raises algebraic questions concerning tensors. Everyone knows that matrices can be diagonalized in some common orthonormal basis if and only if they are symmetric and commute with each other; we shall see an analogous statement for tensors with more
than two indices. This, and other theorems in the same vein, make it possible to associate to any multidimensional Azema martingale an orthogonal decomposition of the state space into one- and two-dimensional subspaces; the behaviour of the process becomes simpler when split into its components in these sub-spaces.

"Rough paths of inhomogeneous degree of smoothness (Pi-rough paths) can be treated as p-rough paths (of homogeneous degree of
smoothness) for a sufficiently large p. The theory of integration with respect to p-rough paths can be applied to prove existence and uniqueness of solutions of differential equations driven by Pi-rough paths. However the required conditions on the one-form determining the differential equation are too strong and can be weakened. The talk proves the existence and uniqueness under weaker conditions and explores some applications of Pi-rough paths

The behaviour of the tail of the distribution of the first passage time over a fixed level has been known for many years, but until recently little was known about the behaviour of the probability mass function or density function. In this talk we describe recent results of Vatutin and Wachtel, Doney, and Doney and Rivero which give such information whenever the random walk or Levy process is asymptotically stable.

Let $X$ be a Poisson point process and $K$ a d-dimensional convex set.
For a point $x \in X$ denote by $v_X(x)$ the Voronoi cell with respect to $X$, and set $$ v_X (K) := \bigcup_{x \in X \cap K } v_X(x) $$ which is the union of all Voronoi cells with center in $K$. We call $v_X(K)$ the Poisson-Voronoi approximation of $K$.
For $K$ a compact convex set the volume difference $V_d(v_X(K))-V_d(K) $ and the symmetric difference $V_d(v_X(K) \triangle K)$ are investigated.
Estimates for the variance and limit theorems are obtained using the chaotic decomposition of these functions in multiple Wiener-Ito integrals