Old Problems, Classical Methods, New Solutions

We use a powerful extension of the classical method heat potentials, recently developed by present author and his collaborators, to solve several significant problems financial mathematics. consider following in detail: (A) calibrating default boundary structural framework constant intensity; (B) calculating probability for representative bank mean-field framework; (C) finding hitting time density an Ornstein-Uhlenbeck process. Several other problems, including pricing American put options optimal mean-reverting trading strategies, are mentioned passing. Besides, two non-financial applications -- supercooled Stefan problem integrate-and-fire neuroscience briefly discussed as well.