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).

Kramkov's optional decomposition theorem

Let ${\displaystyle (\Omega ,{\mathcal {A}},\{{\mathcal {F}}_{t}\},P)}$ be a filtered probability space with the filtration satisfying the usual conditions.

A ${\displaystyle d}$-dimensional process ${\displaystyle X=(X^{1},\dots ,X^{d})}$ is locally bounded if there exist a sequence of stopping times ${\displaystyle (\tau _{n})_{n\geq 1}}$ such that ${\displaystyle \tau _{n}\to \infty }$ almost surely if ${\displaystyle n\to \infty }$ and ${\displaystyle |X_{t}^{i}|\leq n}$ for ${\displaystyle 1\leq i\leq d}$ and ${\displaystyle t\leq \tau _{n}}$.

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