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 $$V$$ with respect to a family of equivalent martingale measures into the form
 * $$V_t=V_0+(H\cdot X)_t-C_t,\quad t\geq 0,$$

where $$C$$ is an adapted (or optional) process.

The theorem is of particular interest for financial mathematics, where the interpretation is: $$V$$ is the wealth process of a trader, $$(H\cdot X)$$ is the gain/loss and $$C$$ 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 $$C$$ 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 $$(\Omega,\mathcal{A},\{\mathcal{F}_t\},P)$$ be a filtered probability space with the filtration satisfying the usual conditions.

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

Statement
Let $$X=(X^1,\dots,X^d)$$ be $$d$$-dimensional càdlàg (or RCLL) process that is locally bounded. Let $$M(X)\neq \emptyset$$ be the space of equivalent local martingale measures for $$X$$ and without loss of generality let us assume $$P\in M(X)$$.

Let $$V$$ be a positive stochastic process then $$V$$ is a $$Q$$-supermartingale for each $$Q\in M(X)$$ if and only if there exist an $$X$$-integrable and predictable process $$H$$ and an adapted increasing process $$C$$ such that
 * $$V_t=V_0 + (H\cdot X)_t-C_t,\quad t\geq 0.$$

Commentary
The statement is still true under change of measure to an equivalent measure.