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 [math]\displaystyle{ V }[/math] with respect to a family of equivalent martingale measures into the form

[math]\displaystyle{ V_t=V_0+(H\cdot X)_t-C_t,\quad t\geq 0, }[/math]

where [math]\displaystyle{ C }[/math] is an adapted (or optional) process.

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

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

Statement

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

Let [math]\displaystyle{ V }[/math] be a positive stochastic process then [math]\displaystyle{ V }[/math] is a [math]\displaystyle{ Q }[/math]-supermartingale for each [math]\displaystyle{ Q\in M(X) }[/math] if and only if there exist an [math]\displaystyle{ X }[/math]-integrable and predictable process [math]\displaystyle{ H }[/math] and an adapted increasing process [math]\displaystyle{ C }[/math] such that

[math]\displaystyle{ V_t=V_0 + (H\cdot X)_t-C_t,\quad t\geq 0. }[/math]

Commentary

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