On the optimal portfolio problem with partial information and related mean field games with relative performance criteria

TL;DR

This paper addresses optimal portfolio selection under partial information, introduces a new approach for solving the problem, and explores mean field game models with explicit solutions in certain cases.

Contribution

It develops a novel method for the single agent problem under partial information and extends it to mean field games with explicit solutions for specific couplings.

Findings

Derived explicit solutions for mean field games with average wealth coupling.

Established regularity of the value function and closed-form optimal processes.

Represented the game value through single agent solutions and a measure-based PDE.

Abstract

We study optimal portfolio choice models in markets with partial information about the stock's drift. We solve the single agent problem for general utilities using a new approach that yields regularity of the value function and closed form expressions for the optimal processes. We consider a N player game in which players interact through the law of peer's wealth and study its mean field limit. This leads to a a game with field equilibrium. We analyze the cases of separable couplings and general utilities, and represent the value of the game as a compilation of the single player problem and a function solving a non local quasilinear pde in the space of measures. We interpret the findings using elements from indifference valuation and arbitrage free pricing, Finally, when the couplings depend only on the average of peer's wealth, we derive explicit solutions and various regularity results for representative cases.