Kramkov's optional decomposition theorem
In probability theory, Kramkov's optional decomposition theorem is a mathematical theorem on the decomposition of a positive supermartingale with respect to a family of equivalent martingale measures into the form
where is an adapted process.
The theorem is of particular interest for financial mathematics, where the interpretation is: is the wealth process of a trader, is the gain/loss and the consumption process.
The theorem was proven in 1994 by Russian mathematician Dmitry Kramkov. The theorem is named after the Doob-Meyer decomposition but unlike there, the process is no longer predictable but only adapted.
Kramkov's optional decomposition theorem
Let be a filtered probability space with the filtration satisfying the usual conditions.A -dimensional process is locally bounded if there exist a sequence of stopping times such that almost surely if and for and.
Statement
Let be -dimensional càdlàg process that is locally bounded. Let be the space of equivalent [local martingale measure]s for and without [loss of generality] let us assume.Let be a positive stochastic process then is a -supermartingale for each if and only if there exist an -integrable and predictable process and an adapted increasing process such that