Chaotic solution for the Black-Scholes equation.

*(English)*Zbl 1238.47051Summary: 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

$${Y}^{s,\tau}:=\left\{u\in C\left((0,\infty )\right):\underset{x\to \infty}{lim}\frac{u\left(x\right)}{1+{x}^{s}}=0,\phantom{\rule{4pt}{0ex}}\underset{x\to 0}{lim}\frac{u\left(x\right)}{1+{x}^{-\tau}}=0\right\}$$

with norm ${\u2225u\u2225}_{{Y}^{s,\tau}}={sup}_{x>0}\left|\frac{u\left(x\right)}{(1+{x}^{s})(1+{x}^{-\tau})}\right|<\infty $, the Black-Scholes semigroup is strongly continuous and chaotic for $s>1$, $\tau \ge 0$ with $s\nu >1$, where $\sqrt{2}\nu $ is the volatility. The proof relies on the Godefroy-Shapiro hypercyclicity criterion [*G. Godefroy* and *J. H. Shapiro*, J. Funct. Anal. 98, No. 2, 229–269 (1991; Zbl 0732.47016)].

##### MSC:

47N10 | Applications of operator theory in optimization, convex analysis, programming, economics |

47D06 | One-parameter semigroups and linear evolution equations |

91G80 | Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) |

35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |

47A16 | Cyclic vectors, hypercyclic and chaotic operators |