This paper shows the existence and uniqueness of the solution of a backward stochastic differential equation inspired from a model for stochastic differential utility in finance theory. We show our results assuming, when possible, no more than the integrability of the terms involved in the equation. We also show the existence and uniqueness of the solution of a backward-forward stochastic differential equation, where the solution depends explicitly on both the past and the future of its own trajectory, under a more restrictive hypothesis on the Lipschitz constant.