Let $a<b$ and let $(X_k)_{k\ge0}$ be a real-valued process. We define an indicator process $(C_k)$ that tracks the phases during which the process is performing an upcrossing from $a$ to $b$.

Set $$ C_0 := \mathbf 1_{\{X_0<a\}}. $$ For all $k\ge1$, define recursively $$ C_k = \mathbf 1_{\{C_{k-1}=1\}}\mathbf 1_{\{X_{k-1}\le b\}} + \mathbf 1_{\{C_{k-1}=0\}}\mathbf 1_{\{X_{k-1}<a\}}. $$

Define also $$ Y_k := \sum_{\ell=1}^k C_\ell (X_\ell - X_{\ell-1}). $$

Interpretation.

• $C_k=1$ indicates that the process is currently in an upcrossing phase, started after going below $a$.
• Once the process exceeds $b$, the upcrossing phase ends and $C_k$ returns to $0$.
• The process $Y_k$ accumulates only the increments of $X$ occurring during these upcrossing phases.

Each completed upcrossing from $a$ to $b$ contributes at least $b-a$ to the sum $Y_n$, since only the increments occurring during the corresponding phase are accumulated.

The only possible negative contribution comes from an unfinished upcrossing at time $n$, when $X_n<a$. This loss is exactly controlled by the term $(X_n-a)^-$.

Summing the contributions of all completed upcrossings and accounting for the final correction yields the stated inequality. ∎

By the upcrossing inequality, for any $a<b$, $$ (b-a)\,\mathbb E[U_n[a,b]] \le \mathbb E[Y_n] + \mathbb E[(X_n-a)^-]. $$

Since $(X_n)$ is a supermartingale, $(Y_n)$ is also a supermartingale with non-positive expectation, hence $\mathbb E[Y_n]\le0$. The $L^1$ boundedness assumption implies that $\sup_n \mathbb E[(X_n-a)^-]<\infty$.

By the monotone convergence theorem, $$ \mathbb E[U_\infty[a,b]]<\infty, $$ which implies that the total number of upcrossings is almost surely finite. Hence, the process cannot oscillate infinitely many times between $a$ and $b$, and $X_n$ converges almost surely. ∎

Intuition. Conditioning with respect to smaller and smaller $\sigma$-fields means losing information. The conditional expectations stabilize and converge to the conditional expectation with respect to the remaining common information.

Define the process $$ X_k := \mathbb E[Z\mid\mathcal G_k], \qquad -n\le k\le -1. $$ This is a martingale bounded in $L^1$.

Similarly as before, we define $C_1=\mathsf{1}_{X_{-n} < a}$ . For all $-n < k \leq -1$, we set $C_k=\mathsf{1}_{C_{k-1}=1} \mathsf{1}_{X_{k-1} \leq b}+ \mathsf{1}_{C_{k-1}=0} \mathsf{1}_{X_{k-1} <a}$. Define also $Y_k=\sum_{\ell=-n}^k C_\ell (X_\ell - X_{\ell-1})$. Then, for the same reasons, the number of upcrossing $U_n[a,b]$ between $-n$ and $-1$ can be bounded as follows: $$ Y_{-1} \geq (b-a) U_n[a,b] - (X_{-1}-a)^- $$

As before, the integrability of $Z$ ensures that the expected number of upcrossings is finite, which implies that $(X_k)$ converges almost surely as $k\to-\infty$.

The limit must coincide with $\mathbb E[Z\mid\mathcal G_{-\infty}]$, which completes the proof. ∎