On the uniqueness of classical solutions of Cauchy problems

TL;DR

This paper establishes a necessary and sufficient condition on the volatility function for the uniqueness of classical solutions to certain Cauchy problems with terminal conditions of at most linear growth.

Contribution

It provides a precise criterion on the volatility function ensuring the uniqueness of solutions, extending previous known results.

Findings

Identifies a necessary and sufficient condition on volatility for solution uniqueness.

Clarifies the relationship between growth conditions and solution existence.

Enhances understanding of well-posedness in Cauchy problems.

Abstract

Given that the terminal condition is of at most linear growth, it is well known that a Cauchy problem admits a unique classical solution when the coefficient multiplying the second derivative (i.e., the volatility) is also a function of at most linear growth. In this note, we give a condition on the volatility that is necessary and sufficient for a Cauchy problem to admit a unique solution.