The finite horizon investor problem with a budget constraint

We study a model that incorporates a budget constraint in a decision making problem. Our goal is to maximize the expected wealth, where in each time period we can either stop the business getting our current wealth or to continue one additional time period and getting a random revenue. We show that when the wealth is scalar, the problem is NP-hard and we provide an FPTAS. However, when the wealth is vector with at least two components the problem cannot be approximated.