[Paper Review] Dynamic Programming Equations for Portfolio Optimization under Partial Information with Expert Opinions

Rüdiger Frey, Ralf Wunderlich|arXiv (Cornell University)|Mar 11, 2013

Stochastic processes and financial applications2 references6 citations

TL;DR

This paper develops dynamic programming equations for optimal portfolio selection in a market with unobserved Markov-modulated drift, incorporating partial information from stock prices and discrete expert signals. By applying stochastic filtering to transform the problem into full information, it establishes viscosity solution techniques and regularization to solve the resulting Hamilton-Jacobi-Bellman equation, yielding a rigorous framework for optimal investment under uncertainty.

ABSTRACT

This paper investigates optimal portfolio strategies in a market where the drift is driven by an unobserved Markov chain. Information on the state of this chain is obtained from stock prices and expert opinions in the form of signals at random discrete time points. As in Frey et al. (2012), Int. J. Theor. Appl. Finance, 15, No. 1, we use stochastic filtering to transform the original problem into an optimization problem under full information where the state variable is the filter for the Markov chain. The dynamic programming equation for this problem is studied with viscosity-solution techniques and with regularization arguments.

Motivation & Objective

• To address optimal portfolio allocation in financial markets where the drift is governed by an unobserved Markov chain.
• To incorporate noisy expert opinions arriving at random discrete times as additional information sources.
• To transform the partially observed problem into a fully observed one using stochastic filtering techniques.
• To derive and analyze the dynamic programming equation for the filtered Markov chain state.
• To establish the existence and uniqueness of solutions to the Hamilton-Jacobi-Bellman equation via viscosity methods.

Proposed method

• Utilizes stochastic filtering to estimate the unobserved Markov chain state based on observed stock prices and expert signals.
• Transforms the original partial information problem into a full information control problem using the filter as the state variable.
• Derives the dynamic programming equation (HJB equation) for the value function in terms of the filtered state.
• Applies viscosity solution techniques to handle the non-smoothness of the value function in the HJB equation.
• Employs regularization arguments to ensure the existence of solutions and to analyze convergence properties.
• Establishes the connection between the optimal control policy and the solution of the HJB equation under the filtered state.

Experimental results

Research questions

• RQ1 How can optimal portfolio strategies be derived when the market drift is governed by an unobserved Markov chain?
• RQ2 What is the impact of discrete expert signals on portfolio optimization under partial information?
• RQ3 How does stochastic filtering enable the transformation of a partially observed problem into a fully observed one?
• RQ4 What mathematical techniques ensure the solvability of the dynamic programming equation in this context?
• RQ5 How do viscosity solutions and regularization contribute to the analysis of the HJB equation in this setting?

Key findings

• The dynamic programming equation for the optimal portfolio problem is successfully derived under partial information by using the filter of the Markov chain as the state variable.
• Viscosity solution techniques are effective in analyzing the HJB equation even when the value function lacks classical differentiability.
• Regularization arguments provide a rigorous path to proving the existence of solutions to the HJB equation in the context of partially observed Markovian markets.
• The framework enables the construction of optimal investment strategies that optimally combine market data and expert signals.
• The solution structure ensures that the optimal portfolio depends only on the current estimate of the Markov state, as provided by the filter.