Optimal consumption and investment with welfare constraints

This paper investigates an optimal consumption and investment problem of an economic agent who faces a welfare constraint: the agent does not accept her expected utility (continuation value) to fall below a certain fixed level regardless of the time and state. This optimisation problem involves an infinite number of constraints. Using a duality approach, we transform infinitely many constraints into a single constraint and define a dual problem, which becomes a two-dimensional singular control problem. The dual problem provides its associated Hamilton–Jacobi–Bellman (HJB) equation with a gradient constraint. Under a general class of utility functions, we obtain an explicit solution to the HJB equation and provide optimal strategies by establishing a duality theorem. As an example, we consider hyperbolic absolute risk aversion (HARA) utility which may incorporate a government subsidy or basic support, and provide its solutions and implications.