Before revision:  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 .
After revision:  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 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 is also guaranteed .