Chaotic solution for the Black-Scholes equation

Abstract:

The Black-Scholes semigroup is studied on spaces of continuous functions on $(0,\infty )$ which may grow at both 0 and at $\infty ,$ which is important since the standard initial value is an unbounded function. We prove that in the Banach spaces \[ Y^{s,\tau }:=\{u\in C((0,\infty )):\;\lim _{x\rightarrow \infty } \frac {u(x)}{1+x^{s}} =0, \; \lim _{x\rightarrow 0}\frac {u(x)}{1+x^{-\tau }} =0\} \] with norm $\left \Vert u\right \Vert _{Y^{s,\tau }}=\underset {x>0}{\sup }\left \vert \frac {u(x)}{(1+x^{s})(1+x^{-\tau })}\right \vert <\infty ,$ the Black-Scholes semigroup is strongly continuous and chaotic for $s>1$, $\tau \geq 0$ with $s\nu >1$, where $\sqrt 2\nu$ is the volatility. The proof relies on the Godefroy-Shapiro hypercyclicity criterion.