Spectrum and asymptotics of the Black–Scholes partial differential equation in (L¹,L^∞)-interpolation spaces

Wolfgang Arendt and Ben de Pagter

Vol. 202 (2002), No. 1, 1–36

DOI: 10.2140/pjm.2002.202.1

Abstract

Let E be an (L¹,L^∞)-interpolation space. Then (T_E(t)f)(x) = f(e^−tx) defines a group on E. It is strongly continuous if and only if E has order continuous norm. In any case, a generator A_E can be associated with T_E. It is shown that its spectrum is the strip {α_E ≤ Reλ ≤α_E}, where α_E and α_E are the Boyd indices of E. The operator B_E = (A_E)² generates a holomorphic semigroup which governs the Black–Scholes partial differential equation u_t = x²u_xx + xu_x, whose well-posedness, spectrum and asymptotics in E are studied.

Milestones

Received: 27 January 2000

Published: 1 January 2002