Doob decomposition theorem

The theorem on the Doob decomposition, named after the American mathematician Joseph L. Doob, in probability theory is a statement about the representation of a stochastic process as a martingale.

Statement

Be a probability space and a filtration. Each of adapted and integrable stochastic process is then represented as, where is a martingale and predictable, ie it applies: is - measurable for all. By fixing this decomposition is unique. Next is monotone increasing if and only if a submartingale is.

Evidence

You defined for

• And

Then applies. The martingale of and predictability of follow directly from the definition.

Uniqueness follows from the fact that the process is both predictable and an Martingal for further such separation. This is only possible if it is constant.

If a submartingale, then all summands are greater than or equal to 0, that is a monotonically increasing process.