Strict classical subsolution on the extended domain R^{K+1}

Construct a strict classical subsolution for the Hamilton–Jacobi–Bellman equation system F(t, x_{k:K+1}, y, V_k, ∂_t V_k, ∂V_k, ∂^2 V_k) = 0 (equation (F)) arising in the goal-based portfolio selection model with mental accounting when the wealth-domain for x_{1:K+1} is extended from [0, ∞)^{K+1} to R^{K+1}. The subsolution should be classical, strict, and compatible with the gradient constraints −θ_i ≤ ∂_{K+1}V_k − ∂_iV_k ≤ λ_i for i = k, …, K, so that it can be used in a comparison principle on the extended domain.

Background

As part of outlining the technical challenges before presenting the proofs, the authors explain that their comparison principle relies on constructing a strict classical subsolution for the HJB system on the original constrained domain [0, ∞)^{K+1}. They note that a natural idea—extending the spatial domain of the wealth variables from [0, ∞)^{K+1} to the full space R^{K+1} to avoid state-constraint difficulties—fails because they could not construct such a strict classical subsolution on the extended domain.

The difficulty is intertwined with the mental-accounting-induced gradient constraints (−θi ≤ ∂{K+1} V_k − ∂_i V_k ≤ λ_i), which require careful handling in the comparison principle. Establishing a strict classical subsolution on R^{K+1} would enable a more straightforward proof by circumventing boundary state constraints, similar in spirit to approaches used in related work without mental-accounting constraints.