Some Time-invariant Stopping Rule Problems

Let X1,X2, . . . be and i.i.d. sequence. We consider three stopping rule problems for stopping the sequence of partial sums, Sn = ∑n 1 Xi, each of which has a timeinvariance for the payoff that allows us to describe the optimal stopping rule in a particularly simple form, depending on one or two parameters. For certain distributions of the Xn, the optimal rules are found explicitly. The three problems are: (1) stopping with payoff equal to the absolute value of the sum with a cost of time, Yn = |Sn| − nc, (2) stopping with payoff equal to the maximum of the partial sums with a cost of time, Yn = max{S0, S1, . . . , Sn} − nc, and (3) deciding when to give up trying to attain a goal or set a record, Yn = I(Sn ≥ a)−nc, or Yn = I(Sn > a)− nc. For each of these problems, the corresponding problems repeated in time, where the objective is to maximize the rate of return, can also be solved.