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 ${(X_t)}_{t\in 𝕋}$ defined on the filtered probability space $(\mathrm{\Omega },ℱ,{(ℱ)}_{t\in 𝕋},\mathbb{P})$. 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 $𝕋$ 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,

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

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

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

$$𝔼[{(X_t^{*})}^2]\le 4𝔼[X_t^2]$$

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 $𝕋={ℝ}_{+}$. 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

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

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,

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

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

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

$${\parallel X_t^{*}\parallel }_p\le \frac{p}{p-1}{\parallel X_t\parallel }_p.$$

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