0.1 Doob Decomposition Theorem
In the theory of [[knowledge-mathematics-analysis-probability-theory-stochastic-processes-stochastic-processes|stochastic processes]] in discrete time, the Doob Decomposition Theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process (or “drift”) starting at zero.
The analogous theorem in the continuous-time case is the [[doob-meyer-decomposition-theorem]].
For integrable discrete-time process \(\{ X_{n} \}_{n\in\mathbb{N}}\) adapted to filtration \(\{ \mathcal{F}_{n} \}_{n\in\mathbb{N}}\) then the Doob decomposition theorem states that we can express \(X_n\) as \[ \begin{align}X_{n}=M_{n}+A_{n}\end{align} \] where \(M_{n}\) is a martingale and \(A_{n}\) is an integrable predictable process, in the sense that \(A_{n}\) is \(\mathcal{F}_{{n-1}}\)-measurable for each \(n>0\). Furthermore, \(X\) is a submartingale if and only if \(\mathbb{E}[X_{n}-X_{{n-1}}|\mathcal{F}_{{n-1}}]\geq 0\) or, equivalently, if \(A\) is increasing a.s.
Proof: For submartingale \(X_{n},~n\geq 0\) define an increasing sequence \(A_{n},~n\geq 0\) by \[A_{n+1} := \sum_{k=0}^n (\mathbb{E}[X_{k+1}|\mathcal{F}_{k}]-X_{k});\quad n\geq 0,~A_{0}:=0,\]and define a martingale \(M_{n},~n \geq 0\) by \[M_{n+1}:=X_{n+1}-A_{n+1}\quad n\geq 0.\]Note that \(M_n\) is \(\mathcal{F}_{n}\)-measurable, \(M_{n+1}-M_{n}=X_{n+1}-\mathbb{E}[X_{n+1}|\mathcal{F}_{n}];~n\geq 1\) and that \(A_{n+1}\) is \(\mathcal{F}_{n}\)-measurable (or predictable), \(n\geq 0\). \(\square\)
Next we will look at how we apply this theorem, or more specifically how we can construct the processes \(M_{n}, A_{n}\) for general process \(X_{n}\).
Consider an adapted process \((X_{n}, \mathcal{F}_{n})\). To construct \((M_{n}, \mathcal{F}_{n})\) and \((A_{n}, \mathcal{F}_{n})\) satisfying the conditions of Doob Decomposition Theorem we apply the following algorithm:
- Constructing \(A_{n}\): Set: \(A_{0}=0\). Define: \(A_{n}=A_{n-1}+\mathbb{E}[(X_{n}-X_{n-1})|\mathcal{F}_{n-1}]\) which is by design predictable.
- Constructing \(M_{n}\): Set \(M_{0}=0\). Define: \(M_{n}=M_{n-1}+(X_{n}-\mathbb{E}[X_{n}|\mathcal{F}_{n-1}])\) which is by design a martingale.