On some non-linear boundary value problems related to a Black--Scholes model with transaction costs

TL;DR

This paper extends the Black--Scholes model by incorporating variable volatility and abstract boundary conditions, establishing the existence of extremal solutions and broadening the model's applicability in financial mathematics.

Contribution

It introduces a generalized framework for Black--Scholes equations with variable volatility and abstract boundary conditions, providing new existence results for extremal solutions.

Findings

Existence of extremal solutions under generalized conditions

Application examples demonstrating the model's flexibility

Broad class of problems addressed with new boundary conditions

Abstract

We deal with some generalizations on a Black--Scholes model arising in financial mathematics. As novelty in this paper, we consider a variable volatility and abstract functional boundary conditions, which allow us to treat a very large class of problems involving Black--Scholes equation. Our main results involve the existence of extremal solutions in presence of lower and upper solutions. Some examples of application are provided too.