Optimal Investment with Costly Expert Opinions

TL;DR

This paper studies an optimal investment problem where an agent can buy costly expert opinions to inform decisions, using advanced filtering and control techniques to characterize and construct optimal strategies.

Contribution

It introduces a novel approach combining filtering theory and stochastic Perron's method to solve a mixed control problem with costly expert opinions.

Findings

Characterized the value function as a viscosity solution of the PDE.

Reduced the problem to an exit time control problem between expert opinion purchases.

Provided conditions for constructing optimal trading and opinion strategies.

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.