# Portfolio optimization for a large investor controlling market sentiment under partial information

## TL;DR

This paper addresses a complex portfolio optimization problem involving unobservable market states and investor control, using filtering and control theory to derive optimal strategies under partial information.

## Contribution

It introduces a novel approach combining filtering and control theory for jump-diffusion models with unobservable states, providing explicit solutions for utility maximization.

## Key findings

- Optimal portfolio strategies depend on the ability to control the unobservable state process.
- The filtering approach effectively reduces the partial information problem to a full information setting.
- Controlling the state process intensity significantly impacts the investor's optimal wealth and strategies.

## Abstract

We consider an investor faced with the utility maximization problem in which the risky asset price process has pure-jump dynamics affected by an unobservable continuous-time finite-state Markov chain, the intensity of which can also be controlled by actions of the investor. Using the classical filtering theory, we reduce this problem with partial information to one with full information and solve it for logarithmic and power utility functions. In particular, we apply control theory for piecewise deterministic Markov processes (PDMP) to our problem and derive the optimality equation for the value function and characterize the value function as the unique viscosity solution of the associated dynamic programming equation. Finally, we provide a toy example, where the unobservable state process is driven by a two-state Markov chain, and discuss how investor's ability to control the intensity of the state process affects the optimal portfolio strategies as well as the optimal wealth under both partial and full information cases.