On the uniqueness of classical solutions of Cauchy problems

@article{Bayraktar2009OnTU,
  title={On the uniqueness of classical solutions of Cauchy problems},
  author={Erhan Bayraktar and Hao Xing},
  journal={arXiv: Analysis of PDEs},
  year={2009}
}
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. 
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