# Kolmogorov’s martingale inequality

###### Theorem (Kolmogorov’s martingale inequality).

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

 $\mathbb{P}\Bigl{(}\max_{0\leq t\leq T}X(t)\geq\alpha\Bigr{)}\leq\frac{\mathbb{% E}[X(T)^{+}]}{\alpha}\,.$

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

###### Proof.

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

 $B=\Bigl{\{}\max_{1\leq i\leq n}X(t_{i})\geq\alpha\Bigr{\}}$

and split $B$ into disjoint parts $B_{i}$, defined by

 $B_{i}=\Bigl{\{}X(t_{j})<\alpha\text{ for all }j

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

Then

 $\displaystyle\mathbb{P}(B)$ $\displaystyle=\sum_{i=1}^{n}\mathbb{E}\bigl{[}1(B_{i})\bigr{]}$ $\displaystyle\leq\sum_{i=1}^{n}\mathbb{E}\left[\frac{X(t_{i})}{\alpha}\,% \mathbf{1}(B_{i})\right]$ definition of $B_{i}$ $\displaystyle\leq\frac{1}{\alpha}\sum_{i=1}^{n}\mathbb{E}\Bigl{[}\mathbb{E}% \bigl{[}X(T)\mid\mathcal{F}_{t_{i}}\bigr{]}\,\mathbf{1}(B_{i})\Bigr{]}$ $X(t)$ is submartingale $\displaystyle=\frac{1}{\alpha}\sum_{i=1}^{n}\mathbb{E}\Bigl{[}\mathbb{E}\bigl{% [}X(T)\,\mathbf{1}(B_{i})\mid\mathcal{F}_{t_{i}}\bigr{]}\Bigr{]}$ $B_{i}$ is $\mathcal{F}_{t_{i}}$-measurable $\displaystyle=\frac{1}{\alpha}\sum_{i=1}^{n}\mathbb{E}\bigl{[}X(T)\,\mathbf{1}% (B_{i})\bigr{]}$ iterated expectation $\displaystyle=\frac{1}{\alpha}\mathbb{E}\bigl{[}X(T)\,\mathbf{1}(B)\bigr{]}$ $\displaystyle\leq\frac{1}{\alpha}\mathbb{E}\bigl{[}X(T)^{+}\,\mathbf{1}(B)% \bigr{]}$ $\displaystyle\leq\frac{1}{\alpha}\mathbb{E}\bigl{[}X(T)^{+}\bigr{]}$ monotonicity.
 $A=\Bigl{\{}\max_{0\leq t\leq T}X(t)\geq\alpha\Bigr{\}}$

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

 $\mathbb{P}\Bigl{(}\max_{0\leq t\leq T}\lvert X(t)-m\rvert\geq\alpha\Bigr{)}% \leq\frac{\operatorname{Var}[X(T)]}{\alpha^{2}}\,.$
###### Proof.

Apply Kolmogorov’s martingale inequality to $(X(t)-m)^{2}$, which is a submartingale by Jensen’s inequality. ∎

Title Kolmogorov’s martingale inequality KolmogorovsMartingaleInequality 2013-03-22 17:20:22 2013-03-22 17:20:22 stevecheng (10074) stevecheng (10074) 8 stevecheng (10074) Theorem  msc 60G44 msc 60G07 Kolmogorov’s submartingale inequality MarkovsInequality DoobsInequalities