Levy Preservation and Associated Properties for f -Divergence Minimal Equivalent Martingale Measures
We study such important properties of f-divergence minimal martingale measure as Levy preservation property, scaling property, invariance in time property for exponential Levy models. We give some useful decomposition for f-divergence minimal martingale m
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Abstract We study such important properties of f -divergence minimal martingale measure as Levy preservation property, scaling property, invariance in time property for exponential Levy models. We give some useful decomposition for f -divergence minimal martingale measures and we answer on the question which form should have f to ensure mentioned properties. We show that f is not necessarily common f -divergence. For common f -divergences, i.e. functions verifying f 00 .x/ D ax ; a > 0; 2 R, we give necessary and sufficient conditions for existence of f -minimal martingale measure. Keywords f -divergence • Exponential Levy models • Minimal martingale measures • Levy preservation property
Mathematics Subject Classification (2010): 60G07, 60G51, 91B24
1 Introduction This article is devoted to some important and exceptional properties of f -divergences. As known, the notion of f -divergence was introduced by Ciszar [3] to measure the difference between two absolutely continuous probability measures by mean of the expectation of some convex function f of their Radon-Nikodym density. More dQ precisely, let f be a convex function and Z D be a Radon-Nikodym density of dP
S. Cawston () L. Vostrikova LAREMA, D´epartement de Math´ematiques, Universit´e d’Angers, 2, Bd Lavoisier, 49045 Angers Cedex 01, France e-mail: [email protected]; [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 9, © Springer-Verlag Berlin Heidelberg 2013
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two measures Q and P , Q P . Supposing that f .Z/ is integrable with respect to P , f -divergence of Q with respect to P is defined as f .QjjP / D EP Œf .Z/: One can remark immediately that this definition cover such important cases p as variation distance when f .x/ D jx 1j, as Hellinger distance when f .x/ D . x 1/2 and Kulback-Leibler information when f .x/ D x ln.x/. Important as notion, f -divergence was studied in a number of books and articles (see for instance [14, 19]) In financial mathematics it is of particular interest to consider measures Q which minimise on the set of all equivalent martingale measures the f -divergence. This fact is related to the introducing and studying so called incomplete models, like exponential Levy models (see [2,6,7,20,22]). In such models contingent claims cannot, in general, be replicated by admissible strategies. Therefore, it is important to determine strategies which are, in a certain sense optimal. Various criteria are used, some of which are linked to risk minimisation (see [9, 25, 26]) and others consisting in maximizing certain utility functions (see [1,11,16]). It has been shown (see [11,18]) that such questions are strongly linked via Fenchel-Legendre transform to dual optimisation problems, namely to f -divergence minimisation on the set of equivalent martingale measures, i.e. the measures Q which are equivalent to the initial physical m
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