0.1 Doob’s Submartingale Inequality
In mathematics, Doob’s martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a submartingale exceeds any given value over a given interval of time.
Informally, Doob’s inequality states that the expected value of the process at some final time controls the probability that a sample path will reach above any particular value beforehand. As the proof uses very direct reasoning, it does not require any restrictive assumptions on the underlying filtration or on the process itself, unlike for many other theorems about stochastic processes. In the continuous-time setting, right-continuity (or left-continuity) of the sample paths is required, but only for the sake of knowing that the supremal value of a sample path equals the supremum over an arbitrary countable dense subset of times.
Suppose that \(X_{n},~n\geq 0\) is a submartingale. Then, for every \(\lambda>0\) and \(n\geq{0}\) we have \[ \lambda\cdot \mathbb{P}\left( \max_{0\leq k\leq n}(X_{k})\geq \lambda \right) \leq \mathbb{E}\left[ X_{n} \cdot \mathbb{1}_{\{ \max_{0\leq k\leq n} X_{k}\geq \lambda \}} \right] \leq \mathbb{E}[X_{n}^+]. \]
Note that after some rearranging the result is very similar in form to the Markov Inequality \[ \mathbb{P}\left( \max_{0\leq k\leq n}(X_{k})\geq \lambda \right)\leq \frac{\mathbb{E}[X_{n}^+]}{\lambda} . \]
\begin{proof} We define a stopping time \[
\tau:=\inf\{k \geq 0:X_{k}\geq \lambda\}
\] (where \(\tau= +\infty\) is \(X_{k}<\lambda\) for all \(k \geq 0\)). Furthermore, we define event \[
A:=\{ \max_{0\leq k\leq n}X_{k}\geq \lambda \}=\{ \tau\leq n \}\in\mathcal{F}_{\tau \wedge n}\subseteq \mathcal{F}_{\tau}.
\] Then, for every \(k=0,1,\dots,n\) we have \(A\subset \{ \tau=k \}\in \mathcal{F}_{k}\) and also \(A \cap \{ \tau\geq n+1 \}\in \mathcal{F}_{n}\). This implies \[
\begin{align}
\mathbb{E}[X_{\tau \wedge n}\cdot \mathbb{1}_{A}] & = \mathbb{E}\left[ \sum_{k=0}^\infty X_{\tau \wedge n}\cdot \mathbb{1}_{A\cap \{ \tau=k \}} \right] =\mathbb{E}\left[ \sum_{k=0}^\infty X_{k}\cdot \mathbb{1}_{A\cap \{ \tau=k \}}+X_{n}\cdot \mathbb{1}_{A \cap \{ \tau \geq n+1 \}} \right] \\
& \leq \mathbb{E}\left[ \sum_{k=0}^n X_{n}\cdot \mathbb{1}_{A\cap \{ \tau=k \}}+X_{n}\cdot \mathbb{1}_{A\cap \{ \tau \geq n+1 \}} \right]=\mathbb{E}[X_{n}\cdot \mathbb{1}_{A}],
\end{align}
\] where the inequality follows from the submartingale property. Using this inequality, since \(X_{\tau}\geq \lambda\) on the set \(A\), we obtain \[
\begin{align}
\lambda \cdot \mathbb{P}\left( \max_{0\leq k\leq n} X_{k}\geq \lambda \right) & = \lambda \cdot \mathbb{P}(A) = \mathbb{E}[\lambda \mathbb{1}_{A}]\leq \mathbb{E}[X_{\tau}\cdot \mathbb{1}_{A}]=\mathbb{E}[X_{\tau \wedge n}\cdot \mathbb{1}_{\tau\leq n}] \\
& \leq \mathbb{E}[X_{n}\cdot \mathbb{1}_{\tau \leq n}] = \mathbb{E}[X_{n}\cdot \mathbb{1}_{\max_{0\leq k\leq n}(X_{k})\geq \lambda}]\leq \mathbb{E}[X_{n}^+].
\end{align}
\] \end{proof}