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The first-contact time ${\tau }_{\ast }\le T$ is a stopping time: the event $\{{\tau }_{\ast }\le t\}$ is the finite union of events $\{U_s=Y_s\}$ for $s\le t$. It is finite because $U_T=Y_T$. On $\{{\tau }_{\ast }>t\}$ we have $U_t>Y_t$, so the maximum defining the Snell envelope selects continuation andThe stopped process $U_{t\wedge {\tau }_{\ast }}$ is therefore a martingale: before stopping its conditional increment is zero and after stopping its increment vanishes. Applying the stopped-sum argument with zero conditional increments givesThe upper bound from part (a) now proves that first contact is an optimal stopping time.

Take the first entry into the stopping region:This set is nonempty because $U_T=Z_T$. Before ${\tau }^{\ast }$, the recursion hasso the stopped process $U_{t\wedge {\tau }^{\ast }}$ is a martingale. Optional sampling and $U_{{\tau }^{\ast }}=Z_{{\tau }^{\ast }}$ giveThus ${\tau }^{\ast }$ is an optimal stopping time.