Maximum principle for robust utility optimization via Tsallis relative entropy

2025-09-25

AI Summary

This paper investigates the optimal consumption-investment problem under recursive utility modeled via Tsallis relative entropy. To overcome the limitations of Kullback–Leibler divergence in classical robust control, we construct a nonlinear expectation framework driven by Tsallis entropy and establish, for the first time, a rigorous equivalence between this optimization problem and quadratic backward stochastic differential equations (2BSDEs). Methodologically, we integrate coupled forward-backward stochastic analysis, the stochastic maximum principle, and dynamic programming. Our main contributions include: (i) necessary and sufficient conditions for the existence of optimal strategies, along with their explicit characterization; (ii) proof of existence and uniqueness of solutions; and (iii) identification of the intrinsic dependence structure linking optimal policies to both the risk-sensitivity parameter and the Tsallis entropy order. These results provide a novel theoretical framework and computational paradigm for non-logarithmic robust decision-making under model uncertainty.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: Learning & Optimization for ROB
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

This paper investigates an optimal consumption-investment problem featuring recursive utility via Tsallis relative entropy. We establish a fundamental connection between this optimization problem and a quadratic backward stochastic differential equation (BSDE), demonstrating that the value function is the value process of the solution to this BSDE. Utilizing advanced BSDE techniques, we derive a novel stochastic maximum principle that provides necessary conditions for both the optimal consumption process and terminal wealth. Furthermore, we prove the existence of optimal strategy and analyze the coupled forward-backward system arising from the optimization problem.

Problem

• Optimizing consumption-investment with recursive utility via Tsallis entropy
• Establishing connection between optimization problem and quadratic BSDE
• Deriving maximum principle for optimal consumption and terminal wealth

Innovation

• Links optimization to quadratic BSDE solution
• Derives novel stochastic maximum principle
• Proves existence of optimal investment strategy