Doob decomposition theorem

In the theory of discrete time stochastic processes, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of any submartingale as the sum of a martingale and an increasing predictable process. The theorem was proved by and is named for J. L. Doob. The analogous theorem for continuous submartingales is the Doob–Meyer decomposition theorem.

The theorem

Any submartingale X_n has a unique decomposition X_n = M_n + A_n where M_n is a martingale and A_n is a predictable, increasing process with A₀ = 0.

References

• Doob, J.L. (1953). Stochastic Processes. Wiley.
• Durrett, Rick (2005). Probability: Theory and Examples (3 ed.). Brooks/Cole. p. 234. ISBN 0-534-42441-4.