Uwe Küchler

Coherent risk measures and good-deal bounds

Abstract. The relation between coherent risk measures, valuation bounds, and certain classes of portfolio optimization problems is established. One of the key results is that coherent risk measures are essentially equivalent to generalized arbitrage bounds, named “good deal bounds” by Cerny and Hodges (1999). The results are economically general in the sense that they work for any cash stream spaces, be it in dynamic trading settings, one-step models, or deterministic cash streams. They are also mathematically general as they work in (possibly infinite-dimensional) linear spaces.The valuation theory presented seems to fill a gap between arbitrage valuation on the one hand and utility maximization (or equilibrium theory) on the other hand. “Coherent” valuation bounds strike a balance in that the bounds can be sharp enough to be useful in the practice of pricing and still be generic, i.e., somewhat independent of personal preferences, in the way many coherent risk measures are somewhat generic.

Strong discrete time approximation of stochastic differential equations with time delay

This paper considers the derivation of weak discrete time approximations for solutions of stochastic differential equations with time delay. These are suitable for Monte Carlo simulation and allow the computation of expectations for funcionals of stochastic delay equations. The suggested approximations converge in a weak sense.

Langevins stochastic differential equation extended by a time-delayed term

The stochastic differential equation is a generalization of Lange-vins equation, which is obtained if b = 0. Necessary and sufficient conditions on a, b and r are given under which a stationary so...

Stock returns and hyperbolic distributions

The four parameter family of hyperbolic distributions fits very well the daily returns of the German stocks that have been included in the DAX during the period 1974 and 1992. Estimators and confidence regions for the hyperbolic parameters are calculated from the empirical data. In particular, skewness and kurtosis can be modelled much better by hyperbolic distributions than by normal distributions. Dates of outliers are identified with economical or political events in the world. It is indicated how the hyperbolic parameters can be used to compare different stocks.

On stationary solutions of delay differential equations driven by a Lévy process

The stochastic delay differential equationis considered, where Z(t) is a process with independent stationary increments and a is a finite signed measure. We obtain necessary and sufficient conditions for the existence of a stationary solution to this equation in terms of a and the Levy measure of Z.

Delay estimation for some stationary diffusion-type processes

In this paper the asymptotic behaviour of the maximum likelihood and Bayesian estimators of a delay parameter is studied. The observed process is supposed to be the solution of a linear stochastic differential equation with one time delay term. It is shown that these estimators are consistent and their limit distributions are described. The behaviour of the estimators is similar to the behaviour of corresponding estimators in change-point problems. The question of asymptotical efficiency is also discussed.

A note on limit theorems for multivariate martingales

Multivariate versions of the law of large numbers and the central limit theorem for martingales are given in a generality that is often necessary when studying statistical inference for stochastic process models. To illustrate the usefulness of the results, we consider estimation for a multi-dimensional Gaussian diffusion, where results on consistency and asymptotic normality of the maximum likelihood estimator are obtained in cases that were not covered by previously published limit theorems. The results are also applied to martingales of a different nature, which are typical of the problems occuring in connection with statistical inference for stochastic delay equations.

Exponential families of stochastic processes and Lévy processes

Abstract It is shown that a class of Levy processes (processes with independent stationary increments) is connected in a natural way to many exponential families of continuous-time stochastic processes. Specifically, the canonical process has independent increments if the family has a nonempty kernel. In many other exponential models the canonical process can be obtained from a Levy process by a stochastic time transformation. The basic observed process need not have independent increments. It can, for instance, be a diffusion-type process or a counting process. We study such properties of exponential families of processes that concern the attached class of Levy processes or can be derived from it. In particular, likelihood theory and maximum likelihood estimation is considered. A thorough discussion is given of exponential families of Levy processes. We also consider the construction of exponential families of processes with a more complicated dynamics by stochastic time transformation of exponential families of Levy processes.

Richard von Mises

Richard von Mises was professor at the University of Berlin from 1920 to 1933. During these thirteen years he advanced, with great vigour, the field of applied mathematics and stochastics. His research contributions cover a most impressive range, and the impact of his ideas can be felt even today.

On sequential parameter estimation for some linear stochastic differential equations with time delay

Let X(·) be a scalar diffusion type process described by the stochastic differential equation with time delay Assume the vector ϑ = (ϑ0, ϑ1, …, ϑ m )′ is unknown. Based on continuous observation of X(·) for every ε > 0 a sequential plan (T ε, ϑε *) is constructed which estimates ϑ with mean square accuracy ε. The limit behaviour of the duration T ε for ε tending to zero is obtained. The properties of almost surely consistency and asymptotic normality of estimators ϑε * are investigated.