Kramkov's optional decomposition theorem

In probability theory, Kramkov's optional decomposition theorem (or just optional decomposition theorem) is a mathematical theorem on the decomposition of a positive supermartingale ${\displaystyle V}$ with respect to a family of equivalent martingale measures into the form

${\displaystyle V_{t}=V_{0}+(H\cdot X)_{t}-C_{t},\quad t\geq 0,}$

where ${\displaystyle C}$ is an adapted (or optional) process.

The theorem is of particular interest for financial mathematics, where the interpretation is: ${\displaystyle V}$ is the wealth process of a trader, ${\displaystyle (H\cdot X)}$ is the gain/loss and ${\displaystyle C}$ the consumption process.

The theorem was proven in 1994 by Russian mathematician Dmitry Kramkov.^[1] The theorem is named after the Doob-Meyer decomposition but unlike there, the process ${\displaystyle C}$ is no longer predictable but only adapted (which, under the condition of the statement, is the same as dealing with an optional process).