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 ${\displaystyle V}${\displaystyle V} with respect to a family of equivalent martingale measures into the form

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

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

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

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

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

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

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

${\displaystyle V_{t}=V_{0}+(H\cdot X)_{t}-C_{t},\quad t\geq 0.}${\displaystyle V_{t}=V_{0}+(H\cdot X)_{t}-C_{t},\quad t\geq 0.}^{[2] [3]}

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

• Probability theorems