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Analysis of continuous strict local martingales via h-transforms

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  • Pal, Soumik
  • Protter, Philip

Abstract

We study strict local martingales via h-transforms, a method which first appeared in work by Delbaen and Schachermayer. We show that strict local martingales arise whenever there is a consistent family of change of measures where the two measures are not equivalent to one another. Several old and new strict local martingales are identified. We treat examples of diffusions with various boundary behavior, size-bias sampling of diffusion paths, and non-colliding diffusions. A multidimensional generalization to conformal strict local martingales is achieved through Kelvin transform. As curious examples of non-standard behavior, we show by various examples that strict local martingales do not behave uniformly when the function (x-K)+ is applied to them. Implications to the recent literature on financial bubbles are discussed.

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  • Pal, Soumik & Protter, Philip, 2010. "Analysis of continuous strict local martingales via h-transforms," Stochastic Processes and their Applications, Elsevier, vol. 120(8), pages 1424-1443, August.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:8:p:1424-1443
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    1. Steven L. Heston & Mark Loewenstein & Gregory A. Willard, 2007. "Options and Bubbles," Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 359-390.
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