A Conjecture Involving Positive Solutions of a Simple Scalar Linear Time-Varying State Equation with Delay
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 conjecture is supported by theoretical analysis for some special cases and simulations.