Uniqueness in Cauchy problems for diffusive real-valued strict local martingales

TL;DR

This paper establishes conditions for the uniqueness of solutions to Cauchy problems associated with diffusive strict local martingales, using smooth functions and weaker Engelbert-Schmidt conditions, with applications to financial models.

Contribution

It introduces new criteria for uniqueness of classical and weak solutions in Cauchy problems for diffusive strict local martingales, extending previous results under weaker conditions.

Findings

Unique classical solutions under local Hölder condition

Unique weak solutions under Engelbert-Schmidt conditions

Applications to quadratic normal volatility models and Bessel process

Abstract

For a real-valued one dimensional diffusive strict local martingale,, we provide a set of smooth functions in which the Cauchy problem has a unique classical solution under a local H\"older condition. Under the weaker Engelbert-Schmidt conditions, we provide a set in which the Cauchy problem has a unique weak solution. We exemplify our results using quadratic normal volatility models and the two dimensional Bessel process.