| Purdue Computational Finance Program | |
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When is the Kramkov Decomposition Predictable?
April 15, 2002
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Abstract:
One of the fundamental decompositions in modern probability theory is the Doob-Meyer decomposition of a supermartingale, into the unique difference of a local martingale and a predictable increasing process. Indeed, Doob first conjectured such a decomposition might hold in order to extend the Ito integral to martingales, and when P. A. Meyer found the right conditions for Doob's conjecture to hold, it heralded a new era for stochastic integration theory, and later for modern Mathematical Finance. In the study of incomplete markets in finance, however, it often arises that a single process is simultaneously a supermartingale for an infinite family of equivalent probability measures. While the Doob-Meyer decomposition will of course hold for each measure separately, simple examples show that it does not hold simultaneously. Nevertheless, D. Kramkov, building on work of El Karoui and Quenez, established that a universal decomposition does indeed hold, at the sacrifice of the condition that the increasing process be predictable. He showed one can take an optional version. Later Follmer, Kabanov, and also again Kramkov improved the original result. It is interesting, however, both intrinsically and for reasons related to finance, to know when one can take the increasing process to be indeed predictable, and not simply optional. This is the subject of the talk, and it is based on joint work with Freddy Delbaen. |
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2002 Purdue University Last Update: Apr 3, 2002 Please send comments and suggestions to the Webmaster. |