Optimal Investment with Costly Expert Opinions

ArXiv ID: 2409.11569 “View on arXiv”

Authors: Unknown

Abstract

We consider the Merton problem of optimizing expected power utility of terminal wealth in the case of an unobservable Markov-modulated drift. What makes the model special is that the agent is allowed to purchase costly expert opinions of varying quality on the current state of the drift, leading to a mixed stochastic control problem with regular and impulse controls involving random consequences. Using ideas from filtering theory, we first embed the original problem with unobservable drift into a full information problem on a larger state space. The value function of the full information problem is characterized as the unique viscosity solution of the dynamic programming PDE. This characterization is achieved by a new variant of the stochastic Perron’s method, which additionally allows us to show that, in between purchases of expert opinions, the problem reduces to an exit time control problem which is known to admit an optimal feedback control. Under the assumption of sufficient regularity of this feedback map, we are able to construct optimal trading and expert opinion strategies.

Keywords: Stochastic Control, Filtering Theory, Partial Information, Expert Opinions, Merton Problem, Equities (Portfolio Management)

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats
  • Why: The paper is highly theoretical, employing advanced stochastic control, filtering theory, viscosity solutions, and the stochastic Perron method, while it lacks any empirical data, backtests, or implementation details, focusing entirely on mathematical proofs and existence results.
  flowchart TD
    A["Research Goal:<br>Optimal Investment with Costly Expert Opinions"] --> B["Methodology:<br>Stochastic Control & Filtering Theory"]
    B --> C["State Space Transformation:<br>Embed into Full Information Problem"]
    C --> D["Computational Process:<br>Solve DP PDE using<br>Stochastic Perron's Method"]
    D --> E["Control Reduction:<br>Identify Exit Time<br>Control Problem"]
    E --> F["Key Findings:<br>Characterize Value Function<br>Construct Optimal Strategies"]