Chen, A. and Vellekoop, M. (2017). Optimal investment and consumption when allowing terminal debt European Journal of Operational Research, 258(1):385--397.

We analyze a dynamic optimization problem which involves the consumption and investment of an investor with constant relative risk aversion for consumption but with a risk aversion for final wealth which does not necessarily imply that terminal wealth must always be positive. We require risk aversion for terminal wealth to be positive but not monotone: there is a point of maximal risk aversion at zero wealth and the investor may continue to consume when wealth is negative. Using dual optimization methods we can derive explicit solutions and we find that the optimal solution differs in a fundamental way from the case where risk aversion is monotone. It turns out that the optimal consumption function is convex and concave at different wealth levels and that the optimal investment strategy may no longer be monotone as a function of the remaining time to invest and consume.