Some time-invariant stopping rule problems

Let X 1, X 2, … be an i.i.d. sequence. We consider three stopping rule problems for stopping the sequence of partial sums each of which has a time-invariance 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 , (2) stopping with payoff equal to the maximum of the partial sums with a cost of time , and (3) deciding when to give up trying to attain a goal or set a record . 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.