Analytical solutions to the backward Kolmogorov PDE via an adiabatic approximation to the Schrödinger PDE

Analytical solutions to the backward Kolmogorov PDE ∂p∂t+b(y,t)22∂2p∂y2+a(y,t)∂p∂y=0 are very dependent on the functional form of b(y,t) and a(y,t). We suggest one solution technique for obtaining analytical solutions via the use of an adiabatic approximation to the Schrödinger PDE. This approximation takes the specific form of a so-called WKB (W = Wentzel [G. Wentzel, Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik, Z. Phys. 38 (1926) 518-529], K = Kramers [H. Kramers, Wellenmechanik und halbzahlige Quantisierung, Z. Phys. 39 (1926) 828-840], B = Brillouin [L. Brillouin, La mécanique ondulatoire de Schrödinger: une méthode générale de résolution par approximations successives, C. R. Acad. Sci. 183 (1926) 24-26]) approximation. We provide for two examples, in financial option pricing, where we show how the proposed approximation could be of use.