Volume 29, Issue 2 -

In this paper, we survey, extend and improve several bounds for the distribution function and the tail probabilities of portfolios, where the dependence structure within the portfolio is completely unknown or only partially known. We present various methods for obtaining bounds based on rearrangements, duality theory, conditional moments and reduction techniques. In particular, we consider the case where only the simple marginal distributions are known, the general overlapping marginals case where certain joint distributions are known and the case of additional restrictions on the dependence structure, as, for example, the restriction to positive dependence. Some of the bounds pose a considerable numerical challenge. We discuss the quality of the bounds and numerical aspects in some examples.

Dynamic risk measures play an important role for the acceptance or non-acceptance of risks in a bank portfolio. Dynamic consistency and weaker versions like conditional and sequential consistency guarantee that acceptability decisions remain consistent in time. An important set of static risk measures are so-called distortion measures. We extend these risk measures to a dynamic setting within the framework of the notions of consistency as above. As a prominent example, we present the Tail-Value-at-Risk (TVaR).

We discuss a general framework of optimal stopping under constraints. We extend the Snell envelope and the characterization of optimal stopping times to our framework with constraints and give a condition which ensures the existence of a right-continuous version. We then study non-constrained stopping problems which are penalized with Lagrange multipliers. We are going to see that this family of problems approximate the expected reward of the constrained problem and allow us to construct optimal stopping times.

We illustrate a new technique for establishing moderate deviation principles for lack-of-fit tests. The method is applied to the problem of comparing two tests using intermediate asymptotic relative efficiency.