Doob decomposition

The theorem about the Doob decomposition , named after the American mathematician Joseph L. Doob , is a statement in probability theory about the representation of a stochastic process as a martingale . Application is, for example, the representation of the quadratic variation process in discrete time.

statement

Be a probability space and a filter . Each stochastic process adapted and integrable can then be represented as , where a martingale and is predictable , i.e. H. the following applies: is - measurable for everyone . With the determination this decomposition is clear. Next is then exactly monotone increasing if a submartingale is. ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle {\mathcal {F}}=({\mathcal {F}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X=M+A}$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle A_{n+1}}$ ${\displaystyle {\mathcal {F}}_{n}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle A_{0}=0}$ ${\displaystyle A}$ ${\displaystyle X}$

proof

Defined for ${\displaystyle n\in \mathbb {N} }$

• ${\displaystyle M_{n}:=X_{0}+\sum _{k=1}^{n}{\bigl (}X_{k}-\mathbb {E} [X_{k}\mid {\mathcal {F}}_{k-1}]{\bigr )}}$ and
• ${\displaystyle A_{n}:=\sum _{k=1}^{n}{\bigl (}\mathbb {E} [X_{k}\mid {\mathcal {F}}_{k-1}]-X_{k-1}{\bigr )},}$

then applies . The martingale property of and the predictability of follow directly from the definition. ${\displaystyle X_{n}=M_{n}+A_{n}}$ ${\displaystyle M}$ ${\displaystyle A}$

The uniqueness follows from the fact that for further such decomposition the process is both predictable and a martingale. But this is only possible if it is constant. ${\displaystyle X=M'+A'}$ ${\displaystyle M-M'=A'-A}$

If is a submartingale, then all summands are greater than or equal to 0, so is a monotonically growing process. ${\displaystyle X}$ ${\displaystyle A_{n}}$ ${\displaystyle A}$

literature

• JL Doob: Stochastic Processes. Wiley, 1953, ISBN 978-0471218135
• Achim Klenke: Probability Theory. Springer, Berlin / Heidelberg 2008, ISBN 978-3-540-76317-8