Spectrum and asymptotics of the Black-Scholes partial differential equation in \((L^ 1,L^ \infty)\)-inter\-polation spaces.
The authors consider a group of operators given by \(T_E(t)f)(x)=f(e^{-t}x)\), \(t\in \mathbb{R}\), \(x>0\) defined on an \((L^1, L^\infty)\)-interpolation space \(E\). They prove that \(T_E\) is a \(C_0\)-group if and only if \(E\) has order continuous norm, consider its generator \(A_Ef(x)=-xf'(x)\) and show that its spectrum coincides with the strip \(\sigma(A_E)=\{\lambda\in \mathbb C: \underline{\alpha}_E\leq \text{Re}\, \lambda\leq \overline{\alpha}_E\}\) where \(\underline{\alpha}_E, \overline{\alpha}_E\) are Boyd indices. They show as well that the relation \(C_E=(I-A_E)^{-1}\) holds for the Cesàro operator \(C_Ef(x)=\frac 1 x\int_0^xf(u)\,du.\) Finally, they study the spectrum of the operator \(B_E=(A_E)^2\) of the form \(B_ef=x^2f''+xf'\) that coincides with the generator of the geometric Brownian motion.
scientific article; zbMATH DE number 1818935