This is joint work with Richard A. Davis (Columbia Statistics) and Johannes Heiny (Copenhagen). In recent years the sample covariance matrix of high-dimensional vectors with iid entries has attracted a lot of attention. A deep theory exists if the entries of the vectors are iid light-tailed; the Tracy-Widom distribution typically appears as weak limit of the largest eigenvalue of the sample covariance matrix. In the heavy-tailed case (assuming infinite 4th moments) the situation changes dramatically. Work by Soshnikov, Auffinger, Ben Arous and Peche shows that the largest eigenvalues are approximated by the points of a suitable nonhomogeneous Poisson process. We follows this line of research. First, we consider a p-dimensional time series with iid heavy-tailed entries where p is any power of the sample size n. The point process of the scaled eigenvalues of the sample covariance matrix converges weakly to a Poisson process. Next, we consider p-dimensional heavy-tailed time series with dependence through time and across the rows. In particular, we consider entries with a linear dependence or a stochastic volatility structure. In this case, the limiting point process is typically a Poisson cluster process. We discuss the suitability of the aforementioned models for large portfolios of return series. 

We describe the possible influence of stochastic 
dependence on the evaluation of
the risk of joint portfolios and establish relevant risk bounds.Some 
basic tools for this purpose are  the distributional transform,the 
rearrangement method and extensions of the classical Hoeffding -Frechet 
bounds based on duality theory.On the other hand these tools find also 
essential applications to various problems of optimal investments,to the 
construction of cost-efficient payoffs as well as to various optimal 
hedging problems.We
discuss in detail the case of optimal payoffs in Levy market models as 
well as utility optimal payoffs and hedgings
with state dependent utilities.

The problem of optimal stopping with finite horizon in discrete time
is considered in view of maximizing the expected gain. The algorithm
presented in this talk is completely nonparametric in the sense that it
uses observed data from the past of the process up to time -n+1 (n being
a natural number), not relying on any specific model assumption. Kernel
regression estimation of conditional expectations and prediction theory
of individual sequences are used as tools.
The main result is that the algorithm is universally consistent: the
achieved expected gain converges to the optimal value for n tending to
infinity, whenever the underlying process is stationary and ergodic.
An application to exercising American options is given.

In this talk, we present a pathwise method to construct confidence 
intervals on the value of some discrete time stochastic dynamic 
programming equations, which arise, e.g., in nonlinear option pricing 
problems such as credit value adjustment and pricing under model 
uncertainty. Our method generalizes the primal-dual approach, which is 
popular and well-studied for Bermudan option pricing problems. In a 
nutshell, the idea is to derive a maximization problem and a 
minimization problem such that the value processes of both problems 
coincide with the solution of the dynamic program and such that 
optimizers can be represented in terms of the solution of the dynamic 
program. Applying an approximate solution to the dynamic program, which 
can be precomputed by any algorithm, then leads to `close-to-optimal' 
controls for these optimization problems and to `tight' lower and upper 
bounds for the value of the dynamic program, provided that the algorithm 
for constructing the approximate solution was `successful'. We 
illustrate the method numerically in the context of credit value 
adjustment and pricing under uncertain volatility.
The talk is based on joint work with C. Gärtner, N. Schweizer, and J. 
Zhuo.