Risk-Sensitive Investment Management via Free Energy-Entropy Duality

2026-04-16

AI Summary

This study addresses the benchmark risk-sensitive portfolio optimization problem by innovatively introducing a free energy–entropy duality, which reformulates the original problem as an entropy-regularized linear–quadratic–Gaussian stochastic differential game under an equivalent measure. The proposed framework unifies the Kuroda–Nagai change-of-measure approach and simultaneously offers dual interpretations: one in terms of fractional Kelly strategies and the other via entropy-regularized Kelly portfolios. Leveraging stochastic control theory, the authors derive a quadratic value function and an affine feedback optimal policy. Empirical validation using U.S. equity market data demonstrates both the computational tractability of the algorithm and the numerical equivalence of the two interpretive pathways.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study a benchmarked risk-sensitive portfolio problem in a factor-based setting to bring together three strands of the literature: benchmarked risk-sensitive investment management, the Kuroda-Nagai change-of-measure method, and the free energy-entropy duality of Dai Pra et al. (1996). We show that the duality yields a direct solution of the benchmarked problem by reformulating it as a linear-quadratic-Gaussian stochastic differential game under a suitable equivalent probability measure, with an entropic regularization. The resulting value function is quadratic, the optimal controls are explicit affine feedback maps, and the optimal allocation admits two complementary interpretations: as a fractional Kelly strategy and as a Kelly portfolio adjusted via the entropic regularization. This formulation, therefore, contributes both a direct analytical route to the solution and a clearer interpretation of risk sensitivity, thereby embedding the classical Kuroda-Nagai change-of-measure approach within a more general framework. An added benefit of this formulation is that it is suitable for implementation via an RL algorithm. A simple implementation on U.S. equity data illustrates the tractability of the framework and numerically confirms the equivalence of the two approaches.

Problem

• risk-sensitive investment
• benchmarked portfolio
• factor-based setting
• stochastic control
• portfolio optimization

Innovation

• free energy-entropy duality
• risk-sensitive portfolio optimization
• stochastic differential game
• entropic regularization
• benchmarking