Monotone operators in mathematical finance: nonlinear black-scholes equation

We treat nonlinear parabolic Cauchy problems for valuation of options in financial markets, especially problems of Black-Scholes-type with nonlinear diffusion. Typically, methods based on viscosity solutions are used for determining the solvability of such fully nonlinear problems. However, the special form of these problems in Financial Mathematics enables us to transform them into abstract initial value problems with monotone second-order differential operators to which classical results for abstract parabolic Cauchy problems can be applied. The transformation from the unknown option price P(S , t) to its partial derivative ∆(S , t) = ∂P ∂S , called the Greek ∆, is very simple. The standard theory of monotone operators in Hilbert spaces (of type L2 with a weight) is applicable to the nonlinear Cauchy problem for the new unknown function ∆(S , t) of the stock price S at time t