Volume 23, Issue 2 - 2

In this paper, we consider the problem of maximizing the expected utility of terminal wealth in the framework of incomplete financial markets. In particular, we analyze the case where an economic agent, who aims at such an optimization, achieves infinite wealth with strictly positive probability. By convex duality theory, this is shown to be equivalent to having the minimal-entropy martingale measure Q ^ non-equivalent to the historical probability P (what we call the absolutely-continuous case ). In this anomalous case, we no longer have the representation of the optimal wealth as the terminal value of a stochastic integral, stated in Schachermayer [9] for the case of Q ^ ∼ P (i.e. the equivalent case ). Nevertheless, we give an approximation of this terminal wealth through solutions to suitably-stopped problems, solutions which still admit the integral representation introduced in [9]. We also provide a class of examples fitting to the absolutely-continuous case.

We propose an estimate for the index of extreme value distribution which based on k n -record values and show its consistency and asymptotic normality. The problem of specifying the optimal value of k  =  k n involved in our estimator is investigated. Some simulation results are also presented in order to illustrate the practical validation of asymptotic results for finite samples.

Stationary multiplier methods are procedures for rounding real probabilities into rational proportions, while the Sainte-Laguë divergence is a reasonable measure for the cumulative error resulting from this rounding step. Assuming the given probabilities to be uniformly distributed, we show that the Sainte-Laguë divergences converge to the Lévy-stable distribution that obtains for the multiplier method with standard rounding. The norming constants to achieve convergence depend in a subtle way on the stationary method used.

We consider sequences of random variables with distributions that satisfy recurrences as they appear for quantities on random trees, random combinatorial structures and recursive algorithms. We study the tails of such random variables in cases where after normalization convergence to the normal distribution holds. General theorems implying subgaussian distributions are derived. Also cases are discussed with non-Gaussian tails. Applications to the probabilistic analysis of algorithms and data structures are given.

We consider a non-parametric regression model with long-range dependent innovations, in which the regression function may have a discontinuity at an unknown point. We propose a method to estimate the unknown time of change. The rate of consistency and limit distribution of the estimator are studied. Monte Carlo simulations are reported to assess the finite sample behavior of the estimator.