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For a paper I am writing, I need these two facts. The proofs are fairly short, but I would rather just cite them. This is for martingales index by natural numbers. Also, I call a martingale which converges to 0 "singular". I have also seen them called "potentials".

  1. Is there a good reference for these two facts?

  2. Do these decompositions have standard names?

  3. Is there a standard term for a martingale which converges to 0?

Below, $\Vert M \Vert$ is the $L^1$-bound of the martingale $M_k$.

Decomposition 1. Let $(M_{k})$ be an $L^{1}$-bounded martingale with respect to the filtration $({\mathcal{F}}_{k})$. Then there are two nonnegative martingales $(P_k)$ and $(N_k)$ such that such that $M_{k}=P_k-N_k$ a.e. for all $k$, and $\left\Vert M\right\Vert =\left\Vert P\right\Vert +\left\Vert N\right\Vert = \Vert P_0 \Vert_1 + \Vert N_0 \Vert_1$. Further, this decomposition is a.e. unique; $(P_k)=\sup_{n\geq k}E[[M_{n}]^{+}\mid\mathcal{F}_{k}]$ a.e.; $N_k=\sup_{n \geq k}E[[M_{n}]^{-}\mid\mathcal{F}_{k}]$ a.e.; $\lim_{k\rightarrow\infty}P_k=[\lim_{k}M_{k}]^{+}$ a.e.; and $\lim_{k\rightarrow\infty}N_k=[\lim_{k}M_{k}]^{-} a.e.$

Decomposition 2. Let $(M_{k})$ be an $L^{1}$-bounded martingale with respect to the filtration $(\mathcal{F}_{k})$ and let $M_{\infty}=\lim_{n}M_{n}$. Then there is a uniformly integrable martingale $(U_k)$ and a singular martingale $(S_k)$ such that $M_{k}=U_k+S_k$ a.e. for all $k$. Further, this decomposition is a.e. unique; $U_k=E[M_{\infty}\mid\mathcal{F}_{k}]$ a.e.; $S_k=E[M_{k}-M_{\infty}\mid\mathcal{F}_{k}]$ a.e.; and $\left\Vert M\right\Vert =\left\Vert U\right\Vert +\left\Vert S\right\Vert $.

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I haven't received an answer or comment yet. Maybe this is not a standard result. If that is the case, I will just prove it in my paper. (It is a paper for logicians so I shouldn't leave it as an exercise for the reader.) –  Jason Rute Mar 15 '12 at 16:51
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1 Answer

up vote 10 down vote accepted

These are the Krickeberg and Riesz decompositions, respectively. A good reference is section 4 of Chapter V in Probabilities and Potential B by Claude Dellacherie and Paul-Andre Meyer.

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Thank you Byron! That chapter seems to be exactly what I need. –  Jason Rute Mar 16 '12 at 20:38
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