Solving Constrained Consumption–Investment Problems by Simulation of Artificial Market Strategies

Utility‐maximizing consumption and investment strategies in closed form are unknown for realistic settings involving portfolio constraints, incomplete markets, and potentially a high number of state variables. Standard numerical methods are hard to implement in such cases. We propose a numerical procedure that combines the abstract idea of artificial, unconstrained complete markets, well‐known closed‐form solutions in affine or quadratic return models, straightforward Monte Carlo simulation, and a standard iterative optimization routine. Our method provides an upper bound on the wealth‐equivalent loss compared to the unknown optimal strategy, and it facilitates our understanding of the economic forces at play by building on closed‐form expressions for the strategies considered. We illustrate and test our method on the life‐cycle problem of an individual who receives unspanned labor income and cannot borrow or short sell. The upper loss bound is small, and our method performs well in comparison with two existing methods.