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 {\displaystyle V} with respect to a family of equivalent martingale measures into the form
V t = V 0 + ( H ⋅ X ) t − C t , t ≥ 0 , {\displaystyle V_{t}=V_{0}+(H\cdot X)_{t}-C_{t},\quad t\geq 0,}
where C {\displaystyle C} is an adapted (or optional) process. The theorem is of particular interest for financial mathematics, where the interpretation is: V {\displaystyle V} is the wealth process of a trader, ( H ⋅ X ) {\displaystyle (H\cdot X)} is the gain/loss and C {\displaystyle 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 {\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 ( Ω , A , { F t } , P ) {\displaystyle (\Omega ,{\mathcal {A}},\{{\mathcal {F}}_{t}\},P)} be a filtered probability space with the filtration satisfying the usual conditions. A d {\displaystyle d} -dimensional process X = ( X 1 , … , X d ) {\displaystyle X=(X^{1},\dots ,X^{d})} is locally bounded if there exist a sequence of stopping times ( τ n ) n ≥ 1 {\displaystyle (\tau _{n})_{n\geq 1}} such that τ n → ∞ {\displaystyle \tau _{n}\to \infty } almost surely if n → ∞ {\displaystyle n\to \infty } and | X t i | ≤ n {\displaystyle |X_{t}^{i}|\leq n} for 1 ≤ i ≤ d {\displaystyle 1\leq i\leq d} and t ≤ τ n {\displaystyle t\leq \tau _{n}} .
Statement Let X = ( X 1 , … , X d ) {\displaystyle X=(X^{1},\dots ,X^{d})} be d {\displaystyle d} -dimensional càdlàg (or RCLL) process that is locally bounded. Let M ( X ) ≠ ∅ {\displaystyle M(X)\neq \emptyset } be the space of equivalent local martingale measures for X {\displaystyle X} and without loss of generality let us assume P ∈ M ( X ) {\displaystyle P\in M(X)} . Let V {\displaystyle V} be a positive stochastic process then V {\displaystyle V} is a Q {\displaystyle Q} -supermartingale for each Q ∈ M ( X ) {\displaystyle Q\in M(X)} if and only if there exist an X {\displaystyle X} -integrable and predictable process H {\displaystyle H} and an adapted increasing process C {\displaystyle C} such that
V t = V 0 + ( H ⋅ X ) t − C t , t ≥ 0. {\displaystyle 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.
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