Doob decomposition theorem

The Doob decomposition theorem is a result in the theory of discrete time stochastic processes that 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 $A_{0}=0$.

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.