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On a construction of majorizing measures on subsets of ℝⁿ with special metrics

100%

Olejnik J.

tom 197

nr 1

1-12

We consider processes Xₜ with values in $L_{p}(Ω,ℱ,P)$ and "time" index t in a subset A of the unit cube. A natural condition of boundedness of increments is assumed. We give a full characterization of the domains A for which all such processes are a.e. continuous. We use the notion of Talagrand's majorizing measure as well as geometrical Paszkiewicz-type characteristics of the set A. A majorizing measure is constructed.

On Paszkiewicz-type criterion for a.e. continuity of processes in $L^{p}$-spaces

100%

Olejnik J.

Banach Center Publications

2010

tom 90

nr 1

103-110

In this paper we consider processes Xₜ with values in $L^{p}$, p ≥ 1 on subsets T of a unit cube in ℝⁿ satisfying a natural condition of boundedness of increments, i.e. a process has bounded increments if for some non-decreasing f: ℝ₊ → ℝ₊ ||Xₜ-Xₛ||ₚ ≤ f(||t-s||), s,t ∈ T. We give a sufficient criterion for a.s. continuity of all processes with bounded increments on subsets of a given set T. This criterion turns out to be necessary for a wide class of functions f. We use a geometrical Paszkiewicz-type characteristic of the set T. Our result generalizes in some way the classical theorem by Kolmogorov.

Arbitrage in a simple model with general transaction costs

100%

Olejnik J.

Applicationes Mathematicae

2005

tom 32

nr 2

177-182

We study a version of no arbitrage condition in a simple model with general transaction costs. Our condition is equivalent to the existence of an equivalent martingale measure.

Reinsurance-a new approach

63%

Paszkiewicz A. , Olejnik J.

Banach Center Publications

2010

tom 90

nr 1

139-151

We describe a new model of multiple reinsurance. The main idea is that the reinsurance premium is paid conditionally. It is motivated by some analysis of the ultimate price of the reinsurance contract. For simplicity we assume that the underlying risk pricing functional is the L₂-norm. An unexpected relation to the general theory of sample regularity of stochastic processes is given.