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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Chaotic solution for the Black-Scholes equation


Authors: Hassan Emamirad, Gisèle Ruiz Goldstein and Jerome A. Goldstein
Journal: Proc. Amer. Math. Soc. 140 (2012), 2043-2052
MSC (2010): Primary 47D06, 91G80, 35Q91
Published electronically: October 5, 2011
MathSciNet review: 2888192
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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

$\displaystyle Y^{s,\tau }:=\{u\in C((0,\infty )):\;\lim _{x\rightarrow \infty }... ...c {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.

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Additional Information

Hassan Emamirad
Affiliation: Laboratoire de Mathématiques, Université de Poitiers, teleport 2, BP 179, 86960 Chassneuil du Poitou, Cedex, France
Email: emamirad@math.univ-poitiers.fr

Gisèle Ruiz Goldstein
Affiliation: Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152
Email: ggoldste@memphis.edu

Jerome A. Goldstein
Affiliation: Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152
Email: jgoldste@memphis.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11069-4
Keywords: Hypercyclic and chaotic semigroup, Black-Scholes equation.
Received by editor(s): August 18, 2009
Received by editor(s) in revised form: September 13, 2010, December 18, 2010, and February 9, 2011
Published electronically: October 5, 2011
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.