Pasting supermartingales with downward jumps ($Y_n=X_n^1{1}_{\{N>n\}}+X_n^2{1}_{\{N\le n\}}$)

If two supermartingales satisfy $X_N^1\ge X_N^2$ on the event that the stopping time $N$ is finite, the displayed pasted adapted process is a supermartingale. On $\{N>n\}$ its next value is at most $X_{n+1}^1$, because a possible switch at $n+1$ only decreases it. On $\{N\le n\}$ it follows the second supermartingale. Conditioning these inequalities proves the assertion. Finite repeated pastings inherit the same property, allowing a multiplicative trading construction for upcrossings.