Robust optimal investment and consumption strategies with portfolio constraints and stochastic environment

We investigate a continuous-time investment–consumption problem with model uncertainty in a general diffusion-based market with random model coefficients. We assume that a power utility investor is ambiguity-averse, with the preference to robustness captured by the homothetic multiplier robust specification, and the investor’s investment and consumption strategies are constrained to closed convex sets. To solve this constrained robust control problem, we employ the stochastic Hamilton–Jacobi–Bellman–Isaacs equations, backward stochastic differential equations, and bounded mean oscillation martingale theory. Furthermore, we show the investor incurs (non-negative) utility loss, i.e. the loss in welfare, if model uncertainty is ignored. When the model coefficients are deterministic, we establish formally the relationship between the investor’s robustness preference and the robust optimal investment–consumption strategy and the value function, and the impact of investment and consumption constraints on the investor’s robust optimal investment–consumption strategy and value function. Extensive numerical experiments highlight the significant impact of ambiguity aversion, consumption and investment constraints, on the investor’s robust optimal investment–consumption strategy, utility loss, and value function. Key findings include: (1) short-selling restriction always reduces the investor’s utility loss when model uncertainty is ignored; (2) the effect of consumption constraints on utility loss is more delicate and relies on the investor’s risk aversion level.