S-shaped Utility Maximization with VaR Constraint and Partial Information

TL;DR

This paper addresses the complex problem of maximizing S-shaped utility under VaR constraints with unobservable drift, providing theoretical insights and practical algorithms for solution implementation.

Contribution

It introduces a semi-closed integral representation for the dual value function and develops three algorithms, including deep neural networks, for solving the constrained utility maximization problem.

Findings

Identifies a critical wealth level for solution feasibility.

Provides a semi-closed form for the dual value function.

Demonstrates the effectiveness of proposed algorithms through numerical examples.

Abstract

We study S-shaped utility maximisation with VaR constraint and unobservable drift coefficient. Using the Bayesian filter, the concavification principle, and the change of measure, we give a semi-closed integral representation for the dual value function and find a critical wealth level that determines if the constrained problem admits a unique optimal solution and Lagrange multiplier or is infeasible. We also propose three algorithms (Lagrange, simulation, deep neural network) to solve the problem and compare their performances with numerical examples.