Kramkov's optional decomposition theorem explained

V with respect to a family of equivalent martingale measures into the formV _{t=V} _{0+(H ⋅} X) _{t-C} _{t,} 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 ⋅ X) is the gain/loss and 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

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, l{A},\{ l{F} _{t\},P)} be a filtered probability space with the filtration satisfying the usual conditions.

A

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

Statement

Let

X = (X ^{1,...,X} ^{d)} be d -dimensional càdlàg (or RCLL) process that is locally bounded. Let M(X) ≠ \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 thatV _{t=V} _{0} + (H ⋅ X) _{t-C} _{t,} t\geq 0 . ^{[2] [3]}

Commentary

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

References

1. Dimitri O.. Kramkov. Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probability Theory and Related Fields. 105. 459–479. 1996. 10.1007/BF01191909. free.
2. Dimitri O.. Kramkov. Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probability Theory and Related Fields. 105. 461. 1996. 10.1007/BF01191909. free.
3. Book: The Mathematics of Arbitrage. Freddy. Delbaen. Walter. Schachermayer. 2006. Springer Berlin. Heidelberg. 31.