# On Positive Solutions of a Delay Equation Arising When Trading in Financial Markets

## TL;DR

This paper analyzes a delay differential equation modeling a trader's account in financial markets, establishing conditions under which the account remains positive or risks bankruptcy, and conjecturing positivity in an intermediate parameter range.

## Contribution

It introduces thresholds for the feedback gain that determine positivity or bankruptcy risk, and provides theoretical and computational support for a conjecture on intermediate cases.

## Key findings

- For  < _-, positivity is guaranteed indefinitely.
- For  > _+, bankruptcy can occur in finite time.
- Support for the conjecture on positivity within the gap _-  _+ is provided.

## Abstract

We consider a discrete-time, linear state equation with delay which arises as a model for a trader's account value when buying and selling a risky asset in a financial market. The state equation includes a nonnegative feedback gain $\alpha$ and a sequence $v(k)$ which models asset returns which are within known bounds but otherwise arbitrary. We introduce two thresholds, $\alpha_-$ and $\alpha_+$, depending on these bounds, and prove that for $\alpha < \alpha_-$, state positivity is guaranteed for all time and all asset-return sequences; i.e., bankruptcy is ruled out and positive solutions of the state equation are continuable indefinitely. On the other hand, for $\alpha > \alpha_+$, we show that there is always a sequence of asset returns for which the state fails to be positive for all time; i.e., along this sequence, bankruptcy is certain and the solution of the state equation ceases to be meaningful after some finite time. Finally, this paper also includes a conjecture which says that for the "gap" interval $\alpha_- \leq \alpha \leq \alpha_+,$ state positivity is also guaranteed for all time. Support for the conjecture, both theoretical and computational, is provided.