Nonlinear PDEs risen when solving some optimization problems in finance, and their solutions

TL;DR

This paper reviews nonlinear PDEs arising in financial optimization problems, discussing their properties and demonstrating numerical and analytical solution methods for specific cases.

Contribution

It provides a comprehensive review of nonlinear PDEs in finance and introduces methods for solving them both numerically and analytically.

Findings

Properties of nonlinear PDEs in finance are characterized.

Numerical solutions can be applied to general cases.

Analytical solutions are feasible for specific cases.

Abstract

We consider a specific type of nonlinear partial differential equations (PDE) that appear in mathematical finance as the result of solving some optimization problems. We review some existing in the literature examples of such problems, and discuss the properties of these PDEs. We also demonstrate how to solve them numerically in a general case, and analytically in some particular case.