11 — Doob’s Upcrossing Lemma and Martingale Convergence
1. Setup
Let $(\Omega,\mathcal{F},P)$ be a probability space and let
${X_n,\mathcal{F}n}{n\ge 0}$ be a submartingale.
Fix two constants
\(a < b.\)
We study how often the process “travels upward” from $a$ to $b$. This leads to upcrossings, a key step in proving the Martingale Convergence Theorem.
2. Upcrossing Stopping Times
Define a sequence of stopping times:
-
Initialization:
\(T_0 = 1.\) -
Down-cross detection: \(T_{2k-1} = \inf\{n > T_{2k-2} : X_n \le a\}, \qquad k\ge 1.\)
-
Up-cross detection: \(T_{2k} = \inf\{n > T_{2k-1} : X_n \ge b\}, \qquad k\ge 1.\)
Define the number of completed upcrossings of the interval $[a,b]$ before time $n$: \(U_n^{a,b} = \sum_{k\ge 1} \mathbf{1}_{\{T_{2k} \le n\}}.\)
3. The Upcrossing Lemma
Durrett’s version states:
Upcrossing Lemma
For any submartingale ${X_n,\mathcal{F}_n}$, \((b-a)\, \mathbb{E}[U_n^{a,b}] \;\le\; \mathbb{E}\big[(X_n-a)^+\big] - \mathbb{E}\big[(X_0-a)^+\big].\)
In particular, \((b-a)\,E[U_n^{a,b}] \;\le\; E[X_n^+] + |a|.\)
This inequality is the engine behind controlling how often the process can oscillate between two levels.
4. Predictable Processes and the Upcrossing Argument
On page 1 and 2 of the lecture notes :contentReference[oaicite:1]{index=1}, the process
\(H_m =
\mathbf{1}_{\{T_{2k} \ge m\}}
-
\mathbf{1}_{\{T_{2k-1} \ge m\}}\)
is introduced.
This $H_m$ is predictable, meaning $H_m$ is measurable with respect to $\mathcal{F}_{m-1}$.
The key transformation:
The stochastic integral
\((H\cdot X)_n = \sum_{m=1}^n H_m (X_m - X_{m-1})\) tracks profit from buying at a downcrossing and selling at an upcrossing.
A clean version of the inequality from the notes:
\[(b-a) U_n^{a,b} \;\le\; (H\cdot X)_n.\]Because for submartingales, \(E[(H\cdot X)_n] \ge 0,\) we obtain the Upcrossing Lemma.
5. Dealing With Down-Crossings
To avoid “re-entry” below $a$, define the truncated process: \(Y_k = a + (X_k - a)^+ = \begin{cases} X_k, & X_k > a, \\ a, & X_k \le a . \end{cases}\)
Then the number of upcrossings of $X$ equals the number of upcrossings of $Y$: \(U_n^{a,b}(X) = U_n^{a,b}(Y).\)
This allows one to stay within the domain where the submartingale property behaves nicely, eliminating issues caused by drops below $a$.
6. Doob’s Martingale Convergence Theorem
The upcrossing lemma gives the main tool for the proof.
Theorem (Doob)
Let ${X_n,\mathcal{F}_n}$ be a submartingale.
If
\(\sup_n E[X_n^+] < \infty,\)
then:
- $X_n \to X$ almost surely,
- $X \in L^1$.
This is shown on page 3 of your notes :contentReference[oaicite:2]{index=2}.
7. Example: Simple Random Walk Stopped at Level 1
Let ${\varepsilon_k}$ be i.i.d. with $P(\varepsilon_k = \pm 1)=1/2$,
\(S_n = \sum_{k=1}^n \varepsilon_k, \qquad S_0 = 0.\)
Let \(T = \inf\{n : S_n = 1\}\) be the hitting time of level 1.
The stopped process ${S_{T\wedge n}}$ is a submartingale with bounded positive part: \(S_{T\wedge n}^+ \le 1.\)
By Doob’s theorem, \(S_{T\wedge n} \xrightarrow{\text{a.s.}} S_T = 1,\) so $T<\infty$ a.s.
However, ${S_{T\wedge n}}$ is not uniformly integrable, hence: \(E(S_{T\wedge n}) = 0 \quad\text{for all }n, \qquad E(S_T)=1.\)
8. Example: Exponential Martingale
Let $Z_k \sim N(0,1)$ i.i.d. and \(S_n = \sum_{k=1}^n Z_k.\)
Define \(Y_n = \exp\left(S_n - \frac{n}{2}\right).\)
Page 2–3 of the notes :contentReference[oaicite:3]{index=3} show:
- ${Y_n}$ is a martingale.
- $Y_n \ge 0$.
- $E[Y_n]=1$ for all $n$.
By the martingale convergence theorem,
\(Y_n \to Y \quad\text{a.s.}\)
But,
\(Y_n = \exp\left(n\left(\frac{S_n}{n}-\frac{1}{2}\right)\right)
\to \exp(-\infty) = 0,\)
since $S_n/n \to 0$ a.s.
Thus the limit is degenerate.
The martingale is not UI, so the limit is not integrable: \(E(Y) = 0 \ne E(Y_n)=1.\)
9. Stopping with a Stopping Time
If $T$ is a stopping time and $H_n = \mathbf{1}_{{n\le T}}$, then \((H\cdot X)_n = X_{T\wedge n} - X_0.\)
This gives a compact way of expressing stopped processes and lets one lift submartingales/supermartingales through stopping times.
Summary
- Upcrossings quantify oscillations between $a$ and $b$.
- Doob’s Upcrossing Lemma controls expected oscillations using submartingale structure.
- This leads directly to Doob’s Martingale Convergence Theorem.
- Examples: stopped random walk, exponential Gaussian martingale.
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