Consider a family of probability measures $\{\mu_y : y \in
\partial D\}$ on a bounded open domain $D\subset R^d$ with smooth
boundary.
For any starting point $x \in D$, we run a
a standard $d$-dimensional Brownian motion $B(t) \in R^d $ until it first
exits $D$ at time $\tau$,
at which time it jumps to a point in the domain $D$ according to the
measure $\mu_{B(\tau)}$ at the exit time,
and starts the Brownian motion afresh. The same evolution is repeated
independently each time the process reaches the boundary.
The resulting diffusion process is called Brownian motion with jump
boundary (BMJ).
The spectral gap of non-self-adjoint generator of BMJ, which describes the
exponential
rate of convergence to the invariant measure, is studied.
The main analytic tool is Fourier transforms with only real zeros.

The two main approaches to modeling defaults, structural and intensity based, will be reviewed. We show that perturbation methods are useful in approximating default probabilities in the context of stochastic volatility models. We then consider the case of many names and we discuss various ways of creating correlation of defaults. In highly-dimensional models, Monte Carlo simulations remain a powerful tool for computing prices of credit derivatives such as CDO's tranches and associated greeks. We propose an interactive particle system approach for computing the small probabilities involved in these financial instruments.

Consider the partition function of a directed polymer in an IID
field. Under some mild assumptions on the field, it is a well-known fact
that the free energy of the polymer is equal to some deterministic constant
for almost every realization of the
field and that the upper tail large deviations is exponential. In this
talk I'll discuss the lower tail large deviations and present a method
for estimating it. As a consequence, I'll show that the lower
tail large deviations exhibits three regimes, determined by the
tail of the negative part of the field. The method applies to other
oriented models and can be adapted to non-oriented models as well. This
work extends the results of a recent paper by Cranston Gautier and
Mountford. A preprint is availabe on www.math.uci.edu/~ibenari

Abstract : We consider a simple random walk on Z
, d > 3. We also consider
a collection of i.i.d. positive and bounded random variables ( V? (x) )x?Z d , which will
serve as a random potential. We study the annealed and quenched cost to perform
long crossings in the random potential ? + ? V? (x), where ? is positive constant
and ? > 0 small enough . These costs are measured by the Lyapounov norms We
prove the equality of the annealed and the quenched norm. We will also discuss the
relation between the Lyapounov norms and the path behavior of the random walk
in the random potential.