On non-uniqueness of solutions to degenerate parabolic equations in the context of option pricing in the Heston model

TL;DR

This paper investigates the non-uniqueness of solutions for degenerate parabolic equations in option pricing models, specifically analyzing the Heston model and demonstrating conditions for uniqueness within certain function classes.

Contribution

It provides a new example confirming the validity of a uniqueness theorem for solutions with sublinear growth in degenerate parabolic equations.

Findings

Demonstrates non-uniqueness in general cases of the Heston model

Constructs a specific example validating the uniqueness theorem

Analyzes the role of function growth conditions in solution uniqueness

Abstract

It is known that the price of call options in the Heston model is determined in a non-unique way. In this paper, this problem is analyzed from the point of view of the existing mathematical theory of uniqueness classes for degenerate parabolic equations. For the special case of degeneracy, a new example is constructed demonstrating the accuracy of the uniqueness theorem for a solution in the class of functions with sublinear growth at infinity.