Solutions to a stationary nonlinear Black-Scholes type equation
The authors study the stationary Dirichlet problem \[ \begin{cases} {1\over 2}\sigma^2S^2{\partial^2 f\over\partial S^2}+{1\over 2} \sigma^2 V^2{\partial^2 f\over\partial\sigma^2}+ \rho\sigma^2 VS{\partial^2f\over \partial S\partial\sigma}- {1\over 2}\rho\sigma^2 V{\partial f\over\partial\sigma}= rg(f)f- rS{\partial f\over\partial S}\quad\text{in }\Omega_0,\\ f= h_0\quad\text{on }\partial\Omega_0.\end{cases}\tag{1a} \] It is motivated by a nonlinear partial differential equation arising in an option pricing model, indeed a Black-Scholes model with stochastic volatilities. After some changing of variables and introducing a parameter \(\lambda\) the new differential equation is \[ \begin{cases} \Delta u+ 2\rho{\partial^2u\over\partial x\partial y}= \lambda(\rho{\partial u\over\partial x}+ (1- {2r\over x^2V^2}){\partial u\over\partial y}+ {2rg(u)\over x^2 V^2}u)\quad\text{in }\Omega,\\ u= h\quad\text{on }\partial\Omega.\end{cases}\tag{1b} \] The authors prove by applying a Newton-type iteration method the existence of a solution in the Sobolev space \(H^2(\Omega)\) under some assumption. A second proof under different assumptions uses the upper and lower solution method.
scientific article; zbMATH DE number 1866743