Before revision:  Sharp results on the existence and uniqueness of classical solutions to the valuation equation are obtained using a combination of probabilistic and analytical techniques. The role of boundary conditions is also discussed .
After revision:  Sharp results on the existence and uniqueness of classical solutions to the valuation equation . 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 .