Vilmos Prokaj

Shadow price in the power utility case

We consider the problem of maximizing expected power utility from consumption over an infinite horizon in the Black-Scholes model with proportional transaction costs, as studied in Shreve and Soner [Ann. Appl. Probab. 4 (1994) 609-692]. Similar to Kallsen and Muhle-Karbe [Ann. Appl. Probab. 20 (2010) 1341-1358], we derive a shadow price, that is, a frictionless price process with values in the bid-ask spread which leads to the same optimal policy.

LOCAL AND TRUE MARTINGALES IN DISCRETE TIME

We prove that for a discrete-time process with infinite time horizon the set of equivalent

Diversity and No Arbitrage

L^p

Stability of a Class of Hybrid Linear Stochastic Systems

-martingale measures is dense in the set of equivalent local martingale measures with respect to the total variation norm.

Some Sufficient Conditions for the Ergodicity of the Lévy Transformation

A stock market is called diverse if no stock can dominate the market in terms of relative capitalization. On one hand, this natural property leads to arbitrage in diffusion models under mild assumptions. On the other hand, it is also easy to construct diffusion models which are both diverse and free of arbitrage. Can one tell whether an observed diverse market admits arbitrage? In this article, we argue that this may well be impossible by proving that the known examples of diverse markets in the literature (which do admit arbitrage) can be approximated uniformly (on the logarithmic scale) by models that are both diverse and arbitrage-free.

ON THE EXTENSIONS OF SUBOPERATORS

The objective of the paper is to establish L q-stability results for the state processes of a new class of slowly time-varying hybrid continuous-time linear stochastic systems. The systems are hybrid in the sense that jumps both in the dynamics and the state may occur. In both cases a jump amounts to resetting to an initial value. Jump-times are defined by a more or less arbitrary point process. Most of the analysis is based on a carefully organized stochastic Lyapunov-function argument. We motivate our study by a brief summary of some basic problems in continuous-time recursive estimation of linear stochastic systems within a framework analogous to the one developed by Benveniste et al.

Séminaire de Probabilités XLVI

We propose a possible way of attacking the question posed originally by Daniel Revuz and Marc Yor in their book published in 1991. They asked whether the Levy transformation of the Wiener-space is ergodic. Our main results are formulated in terms of a strongly stationary sequence of random variables obtained by evaluating the iterated paths at time one. Roughly speaking, this sequence has to approach zero “sufficiently fast”. For example, one of our results states that if the expected hitting time of small neighborhoods of the origin do not grow faster than the inverse of the size of these sets then the Levy transformation is strongly mixing, hence ergodic.

On the Exactness of the Lévy-Transformation

Operator extensions on Hilbert spaces with recent applications to dilation and interpolation problems appeared in [3] while existence theorems are proved in [l-6]. Extremal extensions are described in [4] (in that paper extremal extension means smallest or largest element in a given set of extensions). Here a suboperator means a restriction of a continuous linear operator to a linear (not necessarily closed) subspace of the given Hilbert space. The results in this paper are mainly supplements of the previously known theorems from [l-6]. Nevertheless, as far as the author is aware, they have been remained unnoticed so far. First we give a parametrization of all positive extensions of a suboperator of smallest possible norm with a “unit interval” of operators on an appropriate Hilbert space. This is not really new since it was first proved by Krein [l], that those selfadjoint extensions of a symmetric suboperator, the norm of which does not exceed one, make up an operator interval. Parametrizations of operator intervals similar to ours have appeared in literature before. Then we characterize projection and compact extensions. We are going to use the following construction of Sebestydn [2,3].

Hierarchical Bayesian Modelling of Geographic Dependence of Risk in Household Insurance

This volume provides a broad insight on current, high level researches in probability theory.

Planar diffusions with rank-based characteristics and perturbed Tanaka equations

In a recent paper we gave a sufficient condition for the strong mixing property of the Levy-transformation. In this note we show that it actually implies a much stronger property, namely exactness.