Heun Solutions to the SABR Model

The violation of a uniqueness theorem and an invariant in the application of Poincaré--Perron theorem to Heun's equation

“…Heun's equation appears in mathematical physics problems, in addition to that, it starts to arise in economic and financial problems (SABR model) recently [10,14]. For instance, the Heun functions come out in atomic and nuclear physics (the hydrogen-molecule ion) [32], in the Schrödinger equation with doubly anharmonic potential [22] (its solution is the confluent forms of Heun functions), in the Stark effect [8], in gravitational waves and black holes such as perturbations of the Kerr metric [16,31], in solid state physics (crystalline materials), in Collogero-Moser-Sutherland systems [29], water molecule, graphene electrons [12,23], quantum Rabi model [34], biophysics [26], etc [27,28].…”