Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 39 / 4

Backward induction makes the Snell envelope integrable and adapted: $|U_t|\le |Y_t|+\mathbb{E}(|U_{t+1}|\mid {\mathcal{F}}_t)$. Its definition gives $U_t\ge Y_t$ and $U_t\ge \mathbb{E}(U_{t+1}\mid {\mathcal{F}}_t)$, so it is a supermartingale dominating the reward.

For a stopping time $\tau$ taking values in $\{0,\dots ,T\}$, expand its stopped value asThe indicators are ${\mathcal{F}}_t$-measurable. Taking conditional expectations in each summand makes its expectation nonpositive by the supermartingale property. Since ${\mathcal{F}}_0$ is trivial, $U_0$ is deterministic andThis proves the finite-horizon optional sampling theorem directly in the instance needed here, without assuming nonnegative rewards.

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.

Let $v_j$ be the optimal expected value before seeing the next offer when $j$ rounds remain. With one round left the offer must be accepted, so $v_1=1/2$. For $j>1$, observing $x$ gives a choice between $x$ now and the continuation value $v_{j-1}$. Independence of future uniform distributions makes that continuation value independent of past offers. Thus the uniform-offer stopping recursion isIt gives $v_2=5/8$ and $v_3=89/128$. The Snell envelope rule therefore yields the explicit strategyOnly rounds actually reached are played. At a threshold the two actions have identical continuation expected value; either convention is optimal, and exact equality has probability zero. The optimal expected payout before the first offer is