# Doob’s inequalities

Doob’s inequalities place bounds on the maximum value attained by a martingale in terms of the terminal value. We consider a process $(X_{t})_{t\in\mathbb{T}}$ defined on the filtered probability space $(\Omega,\mathcal{F},(\mathcal{F})_{t\in\mathbb{T}},\mathbb{P})$. The associated maximum process $(X^{*}_{t})$ is

 $X^{*}_{t}\equiv\sup_{s\leq t}|X_{s}|.$

The notation $\|\cdot\|_{p}$ for the $L^{p}$-norm (http://planetmath.org/LpSpace) of a random variable will be used. In discrete-time or, more generally whenever the index set $\mathbb{T}$ is countable, then Doob’s inequalities are as follows.

###### Theorem 1 (Doob).

Let $(X_{t})_{t\in\mathbb{T}}$ be a submartingale with countable index set $\mathbb{T}$. Then,

 $\mathbb{P}\left(\sup_{s\leq t}X_{s}\geq K\right)\leq K^{-1}\mathbb{E}[(X_{t})_% {+}]$ (1)

If $X$ is either a martingale or nonnegative submartingale then,

 $\displaystyle\mathbb{P}(X^{*}_{t}\geq K)\leq K^{-1}\mathbb{E}[|X_{t}|],$ (2) $\displaystyle\|X^{*}_{t}\|_{p}\leq\frac{p}{p-1}\|X_{t}\|_{p}.$ (3)

for every $K>0$ and $p>1$.

In particular, (3) shows that the maximum of any $L^{p}$-bounded martingale is itself $L^{p}$-bounded and, martingales $X^{n}$ converge to $X$ in the $L^{p}$-norm if and only if $(X^{n}-X)^{*}\rightarrow 0$ in the $L^{p}$-norm. The special case where $p=2$ gives

 $\mathbb{E}[(X^{*}_{t})^{2}]\leq 4\mathbb{E}[X_{t}^{2}]$

which is known as Doob’s maximal quadratic inequality.

Similarly, (2) shows that any $L^{1}$-bounded martingale is almost surely bounded and that convergence in the $L^{1}$-norm implies ucp convergence. Inequality (1) is also known as Kolmogorov’s submartingale inequality.

Doob’s inequalities are often applied to continuous-time processes, where $\mathbb{T}=\mathbb{R}_{+}$. In this case, $X^{*}_{t}=\sup_{s\leq t}|X_{s}|$ is a supremum of uncountably many random variables, and need not be measurable. Instead, it is typically assumed that the processes are right-continuous, in which case, for any $t>0$ the supremum may be restricted to the countable set

 $\mathbb{T}^{\prime}=\{s\in\mathbb{R}_{+}:s/t\in\mathbb{Q}\}.$

Putting this into Theorem 1 gives the following continuous-time version of the inequalities.

###### Theorem 2 (Doob).

Let $(X_{t})_{t\in\mathbb{R}_{+}}$ be a right-continuous submartingale. Then,

 $\mathbb{P}\left(\sup_{s\leq t}X_{s}\geq K\right)\leq K^{-1}\mathbb{E}[(X_{t})]$

for every $K>0$. If $X$ is right-continuous and either a martingale or nonnegative submartingale then,

 $\displaystyle\mathbb{P}(X^{*}_{t}\geq K)\leq K^{-1}\mathbb{E}[|X_{t}|],$ $\displaystyle\|X^{*}_{t}\|_{p}\leq\frac{p}{p-1}\|X_{t}\|_{p}.$

for every $K>0$ and $p>1$.

Title Doob’s inequalities DoobsInequalities 2013-03-22 18:39:52 2013-03-22 18:39:52 gel (22282) gel (22282) 7 gel (22282) Theorem msc 60G46 msc 60G44 msc 60G42 Doob’s inequality KolmogorovsMartingaleInequality Doob’s maximal quadratic inequality