Prediction and Doob Decomposition

2012 ◽  
pp. 202-217
Keyword(s):  
Doob Decomposition

Martingales

2021 ◽  
pp. 313-343
Author(s):  
James Davidson
Keyword(s):  
Martingale Theory ◽  
Basic Properties ◽  
Martingale Inequalities ◽  
Martingale Convergence ◽  
Doob Decomposition

This chapter summarizes the essentials of sequential conditioning and martingale theory. After a review with examples of the basic properties of martingales and semi‐martingales, including the Doob decomposition, the upcrossing inequality and martingale convergence are studied and also the role of the conditional variances in establishing convergence. The important martingale inequalities of Kolmogorov, Doob, Burkholder, and Azuma are proved.

scholarly journals NEW MARTINGALE INEQUALITIES IN REARRANGEMENT-INVARIANT FUNCTION SPACES

2004 ◽  
Vol 47 (3) ◽  
pp. 633-657 ◽  
Author(s):  
Masato Kikuchi
Keyword(s):  
Function Space ◽  
Banach Function Space ◽  
Invariant Function ◽  
Secondary 46E30 ◽  
Subject Classification ◽  
Boyd Indices ◽  
Martingale Inequalities ◽  
Rearrangement Invariant Function Spaces ◽  
Doob Decomposition ◽  
L Log L

AbstractWe establish various martingale inequalities in a rearrangement-invariant (RI) Banach function space. If $X$ is an RI space that is not too small, we associate with it RI spaces $\mathcal{H}_p(X)$ $(1\leq p\lt\infty)$ and $K(X)$, and discuss martingale inequalities in these spaces. One of our results is as follows. Let $1\leq p\lt\infty$, let $f=(f_n)$ be an $L_p$-bounded martingale, and let $|f|^p=g+h$ be the Doob decomposition of the submartingale $|f|^p=(|f_n|^p)$ into a martingale $g=(g_n)$ and a predictable non-decreasing process $h=(h_n)$ with $h_0=0$. Then, in the case where $1\ltp\lt\infty$, we obtain the inequalities$$ \|h_{\infty}^{1/p}\|_X\leq2\|f_{\infty}\|_{\mathcal{H}_p(X)}\quad\text{and}\quad \Big\|\sup_n|g_n|^{1/p}\Big\|_X\leq4\|f_{\infty}\|_{\mathcal{H}_p(X)}, $$and, in the case where $p=1$, we obtain the inequalities$$ \|h_{\infty}\|_X\leq\sup_{n\in\mathbb{Z}_{+}}\|f_n\|_{K(X)}\quad\text{and}\quad \sup_{n\in\mathbb{Z}_{+}}\|g_n\|_X\leq2\sup_{n\in\mathbb{Z}_{+}}\|f_n\|_{K(X)}. $$For some specific choices of $X$, we can give explicit expressions for $\mathcal{H}_p(X)$ and $K(X)$. For example, $\mathcal{H}_1(L_1)=L\log L$, $\mathcal{H}_p(L_{p,\infty})=L_{p,1}$, and so on. Furthermore, if the Boyd indices of $X$ satisfy $0\lt\alpha_{X}\leq\beta_{X}\lt1/p$ (respectively, $0\lt\alpha_{X}$), then $\mathcal{H}_p(X)=X$ (respectively, $K(X)=X$). In any case, $\mathcal{H}_p(X)$ is embedded in $K(X)$, and $K(X)$ is embedded in $X$.AMS 2000 Mathematics subject classification: Primary 60G42; 60G46. Secondary 46E30

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