Optimal investments for the standard maximization problem with non-concave utility function in complete market model

We study the standard utility maximization problem for a non-decreasing upper-semicontinuous utility function satisfying mild growth assumption. In contrast to the classical setting, we do not impose the assumption that the utility function is concave. By considering the concave envelope, or concavification, of the utility function, we identify the optimal solution for the optimization problem. We also construct the optimal solution for the constrained optimization problem, where the final endowment is bounded from above by a discrete random variable. We present several examples illustrating that our assumptions cannot be totally avoided.