Deformed martingales and their properties

Abstract

Deformed martingales and their properties

I.V. Pavlov, O.V. Nazarko

Incoming article date: 30.10.2013

With the help of so-called deformations (that is of probability measures families on sigma-fields forming a filtration) a notion of deformed martingale is introduced. This notion generalizes the classical concept of martingale with discrete time. We differ two sorts of deformed martingales: deformed martingales of the 1-st and the 2-nd type. Similarly deformed sub- and supermartingales of the 1-st and the 2-nd type are introduced. We prove that infimum of arbitrary family of deformed supermartingales is a supermartingale and that convexe function of deformed martingale is a deformed submartingale. In addition, for deformed martingales of the 2-nd type a telescopic property is obtained.

Keywords: Filtration, probability measure, deformation, deformed martingale, telescopic property