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

$V_t=V_0+(H\cdot X{)}_t-C_t,t\ge 0,$

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 $(\mathrm{\Omega },{𝒜},\{{{ℱ}}_t\},P)$ 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 $({\tau }_n{)}_{n\ge 1}$ such that ${\tau }_n\to \mathrm{\infty }$ almost surely if $n\to \mathrm{\infty }$ and $|X_t^i|\le n$ for $1\le i\le d$ and $t\le {\tau }_n$.

Statement

Let $X=(X^1,\dots ,X^d)$ be $d$-dimensional càdlàg (or RCLL) process that is locally bounded. Let $M(X)\ne \mathrm{\varnothing }$ 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

$V_t=V_0+(H\cdot X{)}_t-C_t,t\ge 0.$

Commentary

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