VaR-constrained S-shaped utility problem has a critical wealth threshold

What the study found: The study finds a critical wealth level in an S-shaped utility maximization problem with a Value-at-Risk (VaR) constraint and an unobservable drift coefficient. This level determines whether the constrained problem has a unique optimal solution and Lagrange multiplier, or is infeasible.

Why the authors say this matters: The authors suggest that identifying this threshold helps characterize when the constrained optimization problem can be solved. They also propose several numerical approaches to address the problem.

What the researchers tested: The researchers studied S-shaped utility maximization under a VaR constraint when the drift coefficient is not directly observable. They used the Bayesian filter, the concavification principle, and a change of measure to derive a semi-closed integral representation for the dual value function, and they proposed Lagrange, simulation, and deep neural network algorithms.

What worked and what didn't: The paper reports a semi-closed integral representation for the dual value function and a critical wealth level that separates solvable cases from infeasible ones. Numerical examples were used to compare the performance of the three proposed algorithms, but the abstract does not state which method performed best.

What to keep in mind: The abstract does not describe the numerical outcomes in detail or provide the specific comparison results. It also does not state any limitations beyond the model setting with partial information and a VaR constraint.

Key points

• A critical wealth level separates solvable cases from infeasible ones.
• The constrained problem may have a unique optimal solution and Lagrange multiplier when feasible.
• The model involves S-shaped utility, a Value-at-Risk constraint, and an unobservable drift coefficient.
• A semi-closed integral representation was derived for the dual value function.
• Three algorithms were proposed: Lagrange, simulation, and deep neural network.