Reference request: Martingale decompositions (positive/negative and u.i./singular) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T03:56:12Z http://mathoverflow.net/feeds/question/91195 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91195/reference-request-martingale-decompositions-positive-negative-and-u-i-singular Reference request: Martingale decompositions (positive/negative and u.i./singular) Jason Rute 2012-03-14T18:06:16Z 2012-03-16T00:11:46Z <p>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". </p> <blockquote> <ol> <li><p>Is there a good reference for these two facts?</p></li> <li><p>Do these decompositions have standard names?</p></li> <li><p>Is there a standard term for a martingale which converges to 0?</p></li> </ol> </blockquote> <p>Below, $\Vert M \Vert$ is the $L^1$-bound of the martingale $M_k$.</p> <p><strong>Decomposition 1.</strong> Let <code>$(M_{k})$</code> be an $L^{1}$-bounded martingale with respect to the filtration <code>$({\mathcal{F}}_{k})$</code>. Then there are two nonnegative martingales <code>$(P_k)$</code> and <code>$(N_k)$</code> such that such that <code>$M_{k}=P_k-N_k$</code> 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; <code>$(P_k)=\sup_{n\geq k}E[[M_{n}]^{+}\mid\mathcal{F}_{k}]$</code> a.e.; <code>$N_k=\sup_{n \geq k}E[[M_{n}]^{-}\mid\mathcal{F}_{k}]$</code> a.e.; <code>$\lim_{k\rightarrow\infty}P_k=[\lim_{k}M_{k}]^{+}$</code> a.e.; and <code>$\lim_{k\rightarrow\infty}N_k=[\lim_{k}M_{k}]^{-} a.e.$</code></p> <p><strong>Decomposition 2.</strong> Let <code>$(M_{k})$</code> be an $L^{1}$-bounded martingale with respect to the filtration <code>$(\mathcal{F}_{k})$</code> and let <code>$M_{\infty}=\lim_{n}M_{n}$</code>. Then there is a uniformly integrable martingale <code>$(U_k)$</code> and a singular martingale <code>$(S_k)$</code> such that <code>$M_{k}=U_k+S_k$</code> a.e. for all $k$. Further, this decomposition is a.e. unique; <code>$U_k=E[M_{\infty}\mid\mathcal{F}_{k}]$</code> a.e.; <code>$S_k=E[M_{k}-M_{\infty}\mid\mathcal{F}_{k}]$</code> a.e.; and <code>$\left\Vert M\right\Vert =\left\Vert U\right\Vert +\left\Vert S\right\Vert$</code>.</p> http://mathoverflow.net/questions/91195/reference-request-martingale-decompositions-positive-negative-and-u-i-singular/91340#91340 Answer by Byron Schmuland for Reference request: Martingale decompositions (positive/negative and u.i./singular) Byron Schmuland 2012-03-16T00:11:46Z 2012-03-16T00:11:46Z <p>These are the <em>Krickeberg</em> and <em>Riesz</em> decompositions, respectively. A good reference is section 4 of Chapter V in <strong>Probabilities and Potential B</strong> by Claude Dellacherie and Paul-Andre Meyer. </p>