Doob decomposition theorem

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.[1] The analogous theorem for continuous submartingales is the Doob–Meyer decomposition theorem.

The theorem

Any submartingale Xn has a unique decomposition Xn = Mn + An where Mn is a martingale and An is a predictable, increasing process with A0 = 0.[2]

References

  1. ^ Doob, J.L. (1953). Stochastic Processes. Wiley. ISBN 978-0471218135. 
  2. ^ Durrett, Rick (2005). Probability: Theory and Examples (3 ed.). Brooks/Cole. p. 234. ISBN 0-534-42441-4. 
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