# Optimal redeeming strategy of stock loans under drift uncertainty

## TL;DR

This paper investigates the optimal timing for redeeming stock loans when the underlying stock's trend (bull or bear) is uncertain, addressing the challenges posed by incomplete information and degenerate PDEs.

## Contribution

It develops a novel probabilistic and functional analysis to establish regularity of the value function under drift uncertainty, which was previously unaddressed.

## Key findings

- Optimal redeeming strategies differ significantly between bull and bear markets.
- The paper provides a rigorous mathematical framework for handling drift uncertainty in stock loan decisions.
- Regularity results enable better understanding of the value function under incomplete information.

## Abstract

In practice, one must recognize the inevitable incompleteness of information while making decisions. In this paper, we consider the optimal redeeming problem of stock loans under a state of incomplete information presented by the uncertainty in the (bull or bear) trends of the underlying stock. This is called drift uncertainty. Due to the unavoidable need for the estimation of trends while making decisions, the related Hamilton-Jacobi-Bellman (HJB) equation is of a degenerate parabolic type. Hence, it is very hard to obtain its regularity using the standard approach, making the problem different from the existing optimal redeeming problems without drift uncertainty. We present a thorough and delicate probabilistic and functional analysis to obtain the regularity of the value function and the optimal redeeming strategies. The optimal redeeming strategies of stock loans appear significantly different in the bull and bear trends.