Kolmogorov’s martingale inequality
Theorem (Kolmogorov’s martingale inequality).
Let , for , be a submartingale
with continuous sample paths.
Then for any constant ,
(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.
Proof.
Let be a partition of the interval .
Let
and split into disjoint parts , defined by
Also let be the filtration under which
is a submartingale.
Then
definition of | ||||
is submartingale | ||||
is -measurable | ||||
iterated expectation | ||||
monotonicity. |
Since the sample paths are continuous by hypothesis,
the event
can be expressed as an countably infinite intersection
of events of the form with finer and finer partitions
of the time interval .
By taking limits, it follows
has the same bound as the probabilities .
∎
Corollary.
Let , for , be a square-integrable martingale possessing continuous sample paths, whose unconditional mean is . For any constant ,
Proof.
Apply Kolmogorov’s martingale inequality to , which is a submartingale by Jensen’s inequality. ∎
Title | Kolmogorov’s martingale inequality |
---|---|
Canonical name | KolmogorovsMartingaleInequality |
Date of creation | 2013-03-22 17:20:22 |
Last modified on | 2013-03-22 17:20:22 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 8 |
Author | stevecheng (10074) |
Entry type | Theorem![]() |
Classification | msc 60G44 |
Classification | msc 60G07 |
Synonym | Kolmogorov’s submartingale inequality |
Related topic | MarkovsInequality |
Related topic | DoobsInequalities |