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

摘要

Analytical solutions to the backward Kolmogorov PDE[GRAPHICS]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 Schrodinger PDE. This approximation takes the specific form of a so-called WKB (W = Wentzel [G. Wentzel, Eine Verallgemeinerung der Quantenbedingungen fur die Zwecke der Wellenmechanik, Z. Phys. 38 (1926) 518-529], K = Kramers [H. Kramers, Wellenmechanik and halbzahlige Quantisierung, Z. Phys. 39 (1926) 828-840], B = Brillouin [L. Brillouin, La mecanique ondulatoire de Schrodinger: une methode generale de resolution 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. (c) 2005 Elsevier Inc. All rights reserved.