proof of Doob’s inequalities

Let (Ω,,(t)t𝕋,) be a filtered probability space with countable index set 𝕋. If (Xt)t𝕋 is a submartingale, we show that

(supstXsK)K-1𝔼[(Xt)+] (1)

and if X is a martingale or nonnegative submartingale then,

(Xt*K)K-1𝔼[|Xt|], (2)
Xt*ppp-1Xtp. (3)

for every K>0 and p>1.

First, let us consider the case where 𝕋 is finite. The first time at which XtK,


is a stopping time (as hitting times are stopping times). By Doob’s optional sampling theorem for submartingales Xτt𝔼[Xtτt] and therefore,


However, τt if and only if supstXsK giving,

(supstXsK)K-1𝔼[1{supstXsK}Xt], (4)

where the supremum is understood to be over s𝕋. Now suppose that 𝕋 is countable. Then choose finite subsets 𝕋n𝕋 which increase to 𝕋 as n goes to infinity. Replacing 𝕋 by 𝕋n in inequalityMathworldPlanetmath (4) and using the monotone convergence theoremMathworldPlanetmath to take the limit n extends (4) to arbitrary uncountable index sets. Then, inequality (1) follows immediately from (4).

Now, suppose that X is a martingale. Jensen’s inequality gives


for any s<t, so |X| is a nonnegative submartingale. Therefore, it is enough to prove inequalities (2) and (3) for X a nonnegative submartingale, and the martingale case follows by replacing X by |X|.

So, we take X to be a nonnegative submartingale in the following. In this case, (2) just reduces to (1) and it only remains to prove inequality (3).

For p>1, multiply (4) by Kp-1 and integrate up to some limit L>0,

0LKp-1(Xt*K)dK0LKp-2𝔼[1{Xt*K}Xt]dK. (5)

The left hand side of this inequality can be computed by commuting the order of integration with respect to and dK (Fubini’s theorem),


The right hand side of (5) can be computed similarly,


Putting these back into (5),

LXt*pppp-1𝔼[Xt(LXt*)p-1]. (6)

Now let q=p/(p-1), so that p,q are conjugatePlanetmathPlanetmath ( and the Hölder inequality gives


Substituting into (6), the finite term LXt*pp-1 cancels to get


and the result follows by letting L increase to infinity.

Title proof of Doob’s inequalities
Canonical name ProofOfDoobsInequalities
Date of creation 2013-03-22 18:39:55
Last modified on 2013-03-22 18:39:55
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Proof
Classification msc 60G46
Classification msc 60G44
Classification msc 60G42