Nonlinear Parabolic Equations arising in Mathematical Finance

TL;DR

This survey reviews the qualitative and numerical analysis of nonlinear parabolic PDEs in financial mathematics, focusing on extensions of Black-Scholes and stochastic portfolio optimization models.

Contribution

It provides a comprehensive overview of the existence, uniqueness, and numerical schemes for nonlinear parabolic equations in finance, unifying theory and computational methods.

Findings

Existence and uniqueness results for nonlinear parabolic equations.

Stable finite-volume and finite difference schemes for numerical solutions.

Extension of classical models to nonlinear PDE frameworks.

Abstract

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models of stochastic dynamic portfolio optimization leading to the Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both problems can be represented by solutions to nonlinear parabolic equations. Qualitative analysis will be focused on issues concerning the existence and uniqueness of solutions. In the numerical part we discuss a stable finite-volume and finite difference schemes for solving fully nonlinear parabolic equations.