Solvability of a stationary nonlinear Black‐Scholes equation under conditions on the potential - Grossinho, Maria do Rosário (2022)
In this work, we study a nonlinear problem suggested by the Black-Scholes model for option pricing with stochastic volatility, [ vg. Equation p. 129 of this article] where the variables S and <J are respectively the asset value and the market volatility ([l], [2]). In [l], an analogous nonlinear problem has been investigated with a nonlinearity y of the following type y(f) = g(f)f. It has been used an iterative procedure under the hypothesis that g(f)f is nondecreasing, applying upper and lower solutions. The function g was assumed to be C² and this regularity was used in computations in the proofs. We consider a Holder continuous nonlinearity y(f) and, assuming certain conditions on the potential r of y, we prove the existence of a positive solution. The method of the proof, which is based on the construction of upper and lower solutions, obtained as solutions of an auxiliary initial value problem, also yields information on the localization of f.