Advertising for the launch of a product with random success

We discuss the problem of determining the optimal advertising plan for a firm which wants to launch a product, while taking into account that the result of the firm effort will be random rather than deterministic. The firm objectives are to maximize the expected value of the real goodwill level at the launch time and to minimize the total advertising expenditure. The relevant constraint is that of obtaining a successful launch with a probability greater than or equal to a fixed threshold. We observe that, if a special condition on the probability distributions of the final goodwill holds, then the problem is equivalent to a deterministic optimal control problem which has a known optimal solution.