The local quantization behavior of absolutely continuous probabilities

The Existence of Absolutely Continuous Local Martingale Measures

We investigate the existence of an absolutely continuous martingale measure. For continuous processes we show that the absence of arbitrage for general admissible integrands implies the existence of an absolutely continuous (not necessarily equivalent) local martingale measure. We also rephrase Radon-Nikodym theorems for predictable processes. 1.Introduction. In our paper Delbaen and Schacherma...

Computing Tail Probabilities by Numerical Fourier Inversion: the Absolutely Continuous Case

The numerical computation of P{X > x} can be accomplished in a variety of ways. An appealing class of methods may be derived from the contour integral connecting P{X > x} and its Fourier representation. Statisticians have largely focused on deriving saddlepoint approximations for this contour integral. The accuracy of such approximations is generally understood in vague terms only and, perhaps ...

On Absolutely Continuous Transformations

Non-absolutely Continuous Foliations

We consider a partially hyperbolic diffeomorphism of a compact smooth manifold preserving a smooth measure. Assuming that the central distribution is integrable to a foliation with compact smooth leaves we show that this foliation fails to have the absolute continuity property provided that the sum of Lyapunov exponents in the central direction is not zero on a set of positive measure. We also ...

The absolutely continuous spectrum of Jacobi matrices

I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schrödinger operators. These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potentials. In particular, we prove a Denisov-Rakhmanov type theorem for the general finite gap case. The main theme...