Paper: arXiv 2604.15463

Authors: Sebastien Lleo, Wolfgang Runggaldier

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is heavily grounded in advanced stochastic control theory, leveraging free energy-entropy duality and Riccati equations to derive optimal strategies. The empirical component is explicitly described as a mere illustrative proof-of-concept rather than a rigorous backtesting exercise.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Benchmarked Risk-Sensitive Portfolio Problem] --> B[Methodology: Combine Kuroda-Nagai & Free Energy-Entropy Duality]
    B --> C[Input: Factor-based Stochastic Differential Game & US Equity Data]
    C --> D[Process: Entropic Regularization & Change-of-Measure]
    D --> E[Outcome: Explicit Affine Feedback Controls]
    E --> F[Results: Interpretable Kelly-based Strategy & RL Implementability]