A simple conjecture is presented concerning positive solutions of a scalar, time-varying, linear state equation with delay . Although the equation arises in the context of stock trading, no knowledge of finance is needed in the analysis to follow. Starting from positive initial conditions, the state X(k) is governed by a nonnegative constant parameter alpha and a time-varying parameter v(k) with known bounds but otherwise arbitrarily time-varying. We conjecture that if the state X(k) is positive for k less than or equal to N in response to a "distinguished path" v*(k) which we define, then for all admissible paths v(k), positivity of the state  is also guaranteed . The conjectureis supported by theoretical analysis for some special cases and simulations .
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 .
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A simple conjecture is presented concerning positive solutions of a scalar, time-varying,
We consider a discrete-time,
R
. Although the equation arises in the context of stock trading, no knowledge of finance is needed in the analysis to follow. Starting from positive initial conditions, the state X(k) is governed by a nonnegative constant parameter alpha and a time-varying parameter
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
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with
which models asset returns which are within
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arbitrarily time-varying. We conjecture that if the state X(k) is positive for k less than or equal to N in response to a "distinguished path" v*(k) which we define, then for all admissible paths v(k), positivity
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
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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
R
. The conjectureis supported by theoretical analysis for some special cases and simulations
for all time. Support for the conjecture, both theoretical and computational, is provided