Optimal Consumption Until Ruin for an Endowment Described by an Autonomous ODE for an Infinite Time Horizon

Abstract

We give an algorithmic solution of the optimal consumption problem [Formula: see text], where Ct denotes the accumulated consumption until time t, and τ denotes the time of ruin. Moreover, the endowment process Xt is modeled by [Formula: see text]. We solve the problem by showing that the function provided by the algorithm solves the Hamilton-Jacobi (HJ) equation in a viscosity sense and that the same is true for the value function of the problem. The argument is finished by a uniqueness result. It turns out that one has to change the optimal strategy at a sequence of endowment values, described by a free boundary value problem. Finally we give an illustrative example.