202502151902

Tags : Intro to Martingales

Doob Decomposition Theorem

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The Doob decomposition theorem gives a unique decomposition for every adapted process as the sum of a Predicable Process and a Martingale.

Doob's Decomposition Theorem

Let $\mathcal{A}=(\Omega ,\mathcal{F},P)$ be a probability space. $\{{\mathcal{F}}_n\}$ be a filtration of $\mathcal{F}$. Let $X=\{X_n\}$ be an adapted process with $E[[X_i]]<\infty$.

Then there exists a martingale $M$ and an integrable predictable process $A$ starting at $A_0=0$ such that $X_i=M_i+A_i$.

Existence: We have that $M_0=X_0$ and Since $M$ is a martingale, we have that $E[[M_{n+1}-M_n]]=0$ So we can define $M_{n+1}=M_n+(X_{n+1}-E[[X_{n+1}\mid {\mathcal{F}}_n]])$. This is clearly a martingale and we get $A_0=0$ while $A_{n+1}=A_n+(E[[X_{n+1}\mid {\mathcal{F}}_n]]-X_n)$, which is clearly predictable.

Uniqueness: To prove uniqueness, let $X=M+A=M^{\prime }+A^{\prime }$. Then $Y=M-M^{\prime }=A^{\prime }-A$ is a martingale.

But that would mean that $Y$ is both predictable process and a martingale. This would mean that $E[[Y_n|{\mathcal{F}}_n]]=Y_n=Y_{n-1}$ for all $n$. And we have that $Y_0=A_0^{\prime }-A_0=0$ so $Y_n=0$.

This shows the uniqueness.

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References