Analitic approach to solve a degenerate parabolic PDE for the Heston model

TL;DR

This paper develops an analytic method using variational formulation and weighted Sobolev spaces to solve a degenerate parabolic PDE related to the Heston model, ensuring existence and uniqueness of solutions.

Contribution

It introduces a new variational approach with weighted Sobolev spaces for solving degenerate PDEs in the Heston model, establishing existence and uniqueness results.

Findings

Proved existence of weak solutions under certain conditions.

Established uniqueness of solutions for the degenerate PDE.

Provided a framework applicable to unbounded domains in the half-plane.

Abstract

We present an analytic approach to solve a degenerate parabolic problem associated to the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We give a variational formulation, involving weighted Sobolev spaces, of the second order degenerate elliptic operator of the parabolic PDE. We use this approach to prove, under appropriate assumptions on some involved unknown parameters, the existence and uniqueness of weak solutions to the parabolic problem on unbounded subdomains of the half-plane.