Before revision:  It is shown that the value function is the smallest nonnegative classical solution to the valuation equation . Moreover, when the payoff is of strictly sublinear growth, the uniqueness holds among the same class of functions ; when the payoff is of linear growth , the uniqueness holds if and only if the asset-price process is a martingale.
After revision:  It is shown that the value function is the smallest nonnegative classical solution to the valuation equation among the functions with at most linear growth if and only if the asset-price process is a martingale.