Kolmogorov’s martingale inequality
Theorem (Kolmogorov’s martingale inequality).
(The notation means , the positive part of .)
Notice the analogy with Markov’s inequality. Of course, the conclusion is much stronger than Markov’s inequality, as the probabilistic bound applies to an uncountable number of random variables. The continuity and submartingale hypotheses are used to establish the stronger bound.
and split into disjoint parts , defined by
Also let be the filtration under which is a submartingale.
Let , for , be a square-integrable martingale possessing continuous sample paths, whose unconditional mean is . For any constant ,
Apply Kolmogorov’s martingale inequality to , which is a submartingale by Jensen’s inequality. ∎
|Title||Kolmogorov’s martingale inequality|
|Date of creation||2013-03-22 17:20:22|
|Last modified on||2013-03-22 17:20:22|
|Last modified by||stevecheng (10074)|
|Synonym||Kolmogorov’s submartingale inequality|