Abstract :
Analytical solutions to the backward Kolmogorov PDE
∂p
∂t +
b(y, t)2
2
∂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.
2005 Elsevier Inc. All rights reserved
