  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.
