7
$\begingroup$

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 $.

$\endgroup$
1
  • $\begingroup$ 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.) $\endgroup$
    – Jason Rute
    Mar 15, 2012 at 16:51

1 Answer 1

12
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Thank you Byron! That chapter seems to be exactly what I need. $\endgroup$
    – Jason Rute
    Mar 16, 2012 at 20:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.