Paper: arXiv 2501.06275

Abstract

We investigate exploratory randomization for an extended linear-exponential-quadratic-Gaussian (LEQG) control problem in discrete time. This extended control problem is related to the structure of risk-sensitive investment management applications. We introduce exploration through a randomization of the control. Next, we apply the duality between free energy and relative entropy to reduce the LEQG problem to an equivalent risk-neutral LQG control problem with an entropy regularization term, see, e.g. Dai Pra et al. (1996), for which we present a solution approach based on Dynamic Programming. Our approach, based on the energy-entropy duality may also be considered as leading to a justification for the use, in the literature, of an entropy regularization when applying a randomized control.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is highly theoretical, featuring advanced stochastic control theory, duality theorems, and recursive Dynamic Programming solutions without any empirical validation or data. It presents a mathematical framework for exploration in control but lacks backtesting or implementation details.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Explore randomized control for discrete-time<br>Linear-Exponential-Quadratic-Gaussian (LEQG) problem"] --> B["Key Methodology<br>Introduce exploratory randomization<br>Apply free energy & relative entropy duality"]
  B --> C["Mathematical Transformation<br>Reduce LEQG problem to equivalent<br>risk-neutral LQG with entropy regularization"]
  C --> D["Computation<br>Solve via Dynamic Programming<br>obtain optimal state-feedback policy"]
  D --> E["Key Findings/Outcomes<br>1. New optimal control solution for LEQG<br>2. Theoretical justification for entropy regularization<br>3. Application to risk-sensitive investment management"]