Valuation of general contingent claims: Existence,uniqueness, and comparisons of solutions

The Black-Scholes equation for the price u(x,t) of a call option for a single share of common stock with dividend policy d(x,t) is $\text{1}\text{2}\text{σ}{}^2\text{x}{}^2\text{u}{}_{\mathrm{xx}}\text{+(rx?d(x,t))u}{}_{\mathrm{x}}\text{?ru?u}{}_{\mathrm{t}}\text{=0, 0<x, 0<t<T}$, with boundary conditions u(x,0)=max(0,x?E), 0≤x, u(0,t)=0, 0≤t≤T.The coefficients are unbounded and the equation is not uniformly parabolic. We prove an existence (and recall a uniqueness) theorem for a class of equations with boundary conditions that includes the Black-Scholes equation. These may be used to show that an American option must, or will not, sell for the same price as a European option.