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 \begin{equation*} X^*_t\equiv\sup_{s\le t}|X_s|. \end{equation*}The notation $\Vert \cdot\Vert_p$ for the $L^p$ -norm 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, \begin{equation}\label{eq:1} \mathbb{P}\left(\sup_{s\le t}X_s\ge K\right)\le K^{-1}\mathbb{E}[(X_t)_+] \end{equation}If $X$ is either a martingale or nonnegative submartingale then,

	(1)
	(2)

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

In particular, (2) 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 \begin{equation*} \mathbb{E}[(X^*_t)^2]\le 4\mathbb{E}[X_t^2] \end{equation*}which is known as Doob's maximal quadratic inequality.

Similarly, (1) shows that any $L^1$ -bounded martingale is almost surely bounded and that convergence in the $L^1$ -norm implies ucp convergence. Inequality () 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, $X^*_t=\sup_{s\le t}|X_s|$ 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 \begin{equation*} \mathbb{T}^\prime=\{s\in\mathbb{R}_+:s/t\in\mathbb{Q}\}. \end{equation*}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, \begin{equation*} \mathbb{P}\left(\sup_{s\le t}X_s\ge K\right)\le K^{-1}\mathbb{E}[(X_t)] \end{equation*}for every $K>0$ . If $X$ is right-continuous and either a martingale or nonnegative submartingale then,

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