Kolmogorov’s martingale inequality

Theorem (Kolmogorov’s martingale inequality).

Let X(t), for 0tT, be a submartingale with continuousMathworldPlanetmathPlanetmath sample paths. Then for any constant α>0,


(The notation X(T)+ means max(X(T),0), the positive part of X(T).)

Notice the analogyMathworldPlanetmath with Markov’s inequalityMathworldPlanetmath. Of course, the conclusionMathworldPlanetmath is much stronger than Markov’s inequality, as the probabilistic bound applies to an uncountable number of random variablesMathworldPlanetmath. The continuity and submartingale hypotheses are used to establish the stronger bound.


Let {ti}i=1n be a partitionPlanetmathPlanetmath of the interval [0,T]. Let


and split B into disjoint parts Bi, defined by

Bi={X(tj)<α for all j<i but X(ti)α}.

Also let {t} be the filtrationPlanetmathPlanetmath under which X(t) is a submartingale.


(B) =i=1n𝔼[1(Bi)]
i=1n𝔼[X(ti)α 1(Bi)] definition of Bi
1αi=1n𝔼[𝔼[X(T)ti] 1(Bi)] X(t) is submartingale
=1αi=1n𝔼[𝔼[X(T) 1(Bi)ti]] Bi is ti-measurable
=1αi=1n𝔼[X(T) 1(Bi)] iterated expectation
=1α𝔼[X(T) 1(B)]
1α𝔼[X(T)+ 1(B)]
1α𝔼[X(T)+] monotonicity.

Since the sample paths are continuous by hypothesisMathworldPlanetmathPlanetmath, the event


can be expressed as an countably infiniteMathworldPlanetmath intersectionMathworldPlanetmath of events of the form B with finer and finer partitions {ti} of the time interval [0,T]. By taking limits, it follows (A) has the same bound as the probabilities (B). ∎


Let X(t), for 0tT, be a square-integrable martingale possessing continuous sample paths, whose unconditional mean is m=E[X(0)]. For any constant α>0,


Apply Kolmogorov’s martingale inequality to (X(t)-m)2, 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 TheoremMathworldPlanetmath
Classification msc 60G44
Classification msc 60G07
Synonym Kolmogorov’s submartingale inequality
Related topic MarkovsInequality
Related topic DoobsInequalities