Chaotic solution for the Black-Scholes equation

Abstract

The Black-Scholes semigroup is studied on spaces of continuous functions on (0, ∞) which may grow at both 0 and at ∞, which is important since the standard initial value is an unbounded function. We prove that in the Banach spaces Y s,τ := {u ∈ C((0, ∞)) : lim x→∞ u(x) 1 + x s = 0, lim x→0 u(x) 1 + x −τ = 0} with norm u Y s,τ = sup x>0 u(x) (1+x s)(1+x −τ) < ∞, the Black-Scholes semigroup is strongly continuous and chaotic for s > 1, τ ≥ 0 with sν > 1, where √ 2ν is the volatility. The proof relies on the Godefroy-Shapiro hypercyclicity criterion.

A class of quasicontractive semigroups acting on Hardy and weighted Hardy spaces

“…We have proved the following theorem which, as the referee has observed, is formally related to the Black-Scholes equation of mathematical finance (see [8,9]). …”

Strong mixing measures and invariant sets in linear dynamics

“…All the details are proved in Theorem 3.6 in [51]. Thus, the spectral criterion in [45] is satisfied and the Black-Scholes semigroup admits an invariant strongly mixing Borel probability measure on Y s,τ with full support by Corollary 2.2.3.…”

“…Let S s = {λ ∈ C ; 0 < Reλ < sν}. By Lemma 3.5 in [51], we have that g(S s ) ∩ iR = ∅. Then there exists an open ball U ⊂ g(S s ) such that U ∩ iR = ∅ and such that U ∩ R = ∅.…”

“…In [51], it is proved that the Black-Scholes semigroup is strongly continuous and chaotic for s > 1, τ ≥ 0 with sν > 1. We will see that, with a little more work, the Black-Scholes semigroup satisfies the spectral criterion in [45] under the same restrictions on the parameters and, therefore, the hypothesis of Corollary 2.2.3.…”

The specification property in linear dynamics

“…Later, it was proved in [48] that the Black-Scholes semigroup is strongly continuous and chaotic for s > 1, τ ≥ 0 with sν > 1 and it was showed in [79] that it satises the spectral criterion in [42] under the same restrictions on the parameters and, therefore, the hypothesis of Corollary 2.3 of [74] and, consequently, the Black-Scholes semigroup has the SgSP.…”