American Option Prices as Unique Viscosity Solutions to Degenerated Hamilton-Jacobi-Bellman Equations

In this paper we show that the American price of standard (bounded) options in the Black-Scholes one-dimensional model, which is classically given by the value function of an optimal stopping problem, is also the value function of a degenerate stochastic control problem. As a byproduct we get that the American price $u^{*}$ is the unique bounded and continuous viscosity solution of the fully non-linear parabolic equation $-\frac{% \partial u^{*}}{\partial t}\left( t,x\right) =\left( Au^{*}\right) ^{+}\left( t,x\right),t~0, u^{*}\left( T,x\right) =\varphi \left( x\right) $ where $A$ is the infinitesimal generator of the Black-Scholes model, $T$ the maturity and $\varphi $ the payoff of the option