Kolmogorov’s martingale inequality

Theorem (Kolmogorov’s martingale inequality).

Let $X\mathit{}\mathrm{(}t\mathrm{)}$, for $\mathrm{0}\mathrm{\le }t\mathrm{\le }T$, be a submartingale with continuous sample paths. Then for any constant $\alpha \mathrm{>}\mathrm{0}$,

$$\mathbb{P}(\underset{0\le t\le T}{\max }X(t)\ge \alpha )\le \frac{𝔼[X{(T)}^{+}]}{\alpha }.$$

(The notation $X{(T)}^{+}$ means $\max (X(T),0)$, the positive part of $X(T)$.)

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 ${\{t_i\}}_{i=1}^n$ be a partition of the interval $[0,T]$. Let

$$B=\{\underset{1\le i\le n}{\max }X(t_i)\ge \alpha \}$$

and split $B$ into disjoint parts $B_i$, defined by

Also let $\{{ℱ}_t\}$ be the filtration under which $X(t)$ is a submartingale.

Then

$\mathbb{P}(B)$ $={\displaystyle \sum _{i=1}^n}𝔼\left[1(B_i)\right]$

$\le {\displaystyle \sum _{i=1}^n}𝔼\left[{\displaystyle \frac{X(t_i)}{\alpha }}\mathbf{\hspace{0.17em}1}(B_i)\right]$ definition of $B_i$

$\le {\displaystyle \frac{1}{\alpha }}{\displaystyle \sum _{i=1}^n}𝔼\left[𝔼[X(T)\mid {ℱ}_{t_i}]\mathbf{\hspace{0.17em}1}(B_i)\right]$ $X(t)$ is submartingale

$={\displaystyle \frac{1}{\alpha }}{\displaystyle \sum _{i=1}^n}𝔼\left[𝔼[X(T)\mathbf{\hspace{0.17em}1}(B_i)\mid {ℱ}_{t_i}]\right]$ $B_i$ is ${ℱ}_{t_i}$-measurable

$={\displaystyle \frac{1}{\alpha }}{\displaystyle \sum _{i=1}^n}𝔼\left[X(T)\mathbf{\hspace{0.17em}1}(B_i)\right]$ iterated expectation

$={\displaystyle \frac{1}{\alpha }}𝔼\left[X(T)\mathbf{\hspace{0.17em}1}(B)\right]$

$\le {\displaystyle \frac{1}{\alpha }}𝔼\left[X{(T)}^{+}\mathbf{\hspace{0.17em}1}(B)\right]$

$\le {\displaystyle \frac{1}{\alpha }}𝔼\left[X{(T)}^{+}\right]$ monotonicity.

Since the sample paths are continuous by hypothesis, the event

$$A=\{\underset{0\le t\le T}{\max }X(t)\ge \alpha \}$$

can be expressed as an countably infinite intersection of events of the form $B$ with finer and finer partitions $\{t_i\}$ of the time interval $[0,T]$. By taking limits, it follows $\mathbb{P}(A)$ has the same bound as the probabilities $\mathbb{P}(B)$. ∎

Corollary.

Let $X\mathit{}\mathrm{(}t\mathrm{)}$, for $\mathrm{0}\mathrm{\le }t\mathrm{\le }T$, be a square-integrable martingale possessing continuous sample paths, whose unconditional mean is $m\mathrm{=}\mathrm{E}\mathit{}\mathrm{[}X\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{]}$. For any constant $\alpha \mathrm{>}\mathrm{0}$,

$$\mathbb{P}(\underset{0\le t\le T}{\max }|X(t)-m|\ge \alpha )\le \frac{\mathrm{Var}[X(T)]}{{\alpha }^2}.$$

Proof.

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	Theorem
Classification	msc 60G44
Classification	msc 60G07
Synonym	Kolmogorov’s submartingale inequality
Related topic	MarkovsInequality
Related topic	DoobsInequalities