Backward SDEs for Control with Partial Information

TL;DR

This paper addresses a complex non-Markov control problem in finance with hidden factors, proposing a novel approach using backward stochastic differential equations (BSDEs) and duality to overcome the challenges posed by infinite-dimensional filtering distributions.

Contribution

It introduces a new method employing BSDEs and duality to solve non-Markov control problems with partial information, where traditional HJB approaches fail.

Findings

Demonstrates how to formulate the control problem using BSDEs.

Shows the effectiveness of dual formulation in handling infinite-dimensional filtering.

Provides a framework for solving non-Markov control problems in finance.

Abstract

This paper considers a non-Markov control problem arising in a financial market where asset returns depend on hidden factors. The problem is non-Markov because nonlinear filtering is required to make inference on these factors, and hence the associated dynamic program effectively takes the filtering distribution as one of its state variables. This is of significant difficulty because the filtering distribution is a stochastic probability measure of infinite dimension, and therefore the dynamic program has a state that cannot be differentiated in the traditional sense. This lack of differentiability means that the problem cannot be solved using a Hamilton-Jacobi-Bellman (HJB) equation. This paper will show how the problem can be analyzed and solved using backward stochastic differential equations (BSDEs), with a key tool being the problem's dual formulation.