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Let E be an (L 1 ,L∞)-interpolation space. Then (T E (t)f)(x) = f(e -t x) 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 ≤ Re λ ≤ α E }, where α E and α E are the Boyd indices of E. The operator BE = (A E ) 2 generates a holomorphic semigroup which governs the Black-Scholes partial differential equation u t = x 2 u xx + xu x , whose well-posedness, spectrum and asymptotics in E are studied.

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

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