# 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 $(\mathrm{\Omega},\mathcal{F},{(\mathcal{F})}_{t\in \mathbb{T}},\mathbb{P})$. The associated maximum process $({X}_{t}^{*})$ is

$${X}_{t}^{*}\equiv \underset{s\le t}{sup}|{X}_{s}|.$$ |

The notation $\parallel \cdot {\parallel}_{p}$ 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 ${\mathrm{(}{X}_{t}\mathrm{)}}_{t\mathrm{\in}\mathrm{T}}$ be a submartingale with countable index set $\mathrm{T}$. Then,

$$\mathbb{P}(\underset{s\le t}{sup}{X}_{s}\ge K)\le {K}^{-1}\mathbb{E}[{({X}_{t})}_{+}]$$ | (1) |

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

$$\mathbb{P}({X}_{t}^{*}\ge K)\le {K}^{-1}\mathbb{E}[|{X}_{t}|],$$ | (2) | ||

$${\parallel {X}_{t}^{*}\parallel}_{p}\le \frac{p}{p-1}{\parallel {X}_{t}\parallel}_{p}.$$ | (3) |

for every $K\mathrm{>}\mathrm{0}$ and $p\mathrm{>}\mathrm{1}$.

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 ${({X}^{n}-X)}^{*}\to 0$ in the ${L}^{p}$-norm. The special case where $p=2$ gives

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

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, ${X}_{t}^{*}={sup}_{s\le t}|{X}_{s}|$ 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

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

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

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

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

$$\mathbb{P}(\underset{s\le t}{sup}{X}_{s}\ge K)\le {K}^{-1}\mathbb{E}[({X}_{t})]$$ |

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

$$\mathbb{P}({X}_{t}^{*}\ge K)\le {K}^{-1}\mathbb{E}[|{X}_{t}|],$$ | ||

$${\parallel {X}_{t}^{*}\parallel}_{p}\le \frac{p}{p-1}{\parallel {X}_{t}\parallel}_{p}.$$ |

for every $K\mathrm{>}\mathrm{0}$ and $p\mathrm{>}\mathrm{1}$.

Title | Doob’s inequalities |
---|---|

Canonical name | DoobsInequalities |

Date of creation | 2013-03-22 18:39:52 |

Last modified on | 2013-03-22 18:39:52 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 7 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 60G46 |

Classification | msc 60G44 |

Classification | msc 60G42 |

Synonym | Doob’s inequality |

Related topic | KolmogorovsMartingaleInequality |

Defines | Doob’s maximal quadratic inequality |