  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.
