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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
Corrigendum: Proc. Amer. Math. Soc. 142 (2014), 4385-4386
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.

References [Enhancements On Off] (What's this?)

  • [AdP] W. Arendt and B. de Pagter, Spectrum and asymptotics of the Black-Scholes partial differential equation $ (L^1,L^\infty )$-interpolation spaces. Pacific J. Math. 202 (2002), 1-36. MR 1883968 (2002m:35087)
  • [B-S] F. Black and M. Scholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81 (1973), 637-654.
  • [C-G1] D. I. Cruz-Báez and J.M. González-Rodríguez, Semigroup theory applied to options.
    J. Appl. Math. 2 (2002), 131-139. MR 1915662 (2003i:47042)
  • [C-G2] D. I. Cruz-Báez and J.M. González-Rodríguez, A semigroup approach to American
    options.J. Math. Anal. Appl. 302 (2005), 157-165. MR 2107354 (2006b:91066)
  • [deL] R. deLaubenfels, Polynomials of generators of integrated semigroups. Proc. Amer. Math. Soc.107 (1989), 197-204. MR 975637 (90a:47100)
  • [dL-E] R. deLaubenfels and H. Emamirad, Chaos for functions of discrete and continuous weighted shift operators. Ergodic Theory Dynam. Systems.21 (2001), 1411-1427. MR 1855839 (2002j:47030)
  • [DSW] W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators. Ergodic Theory Dynam. Systems 17 (1997), 793-819. MR 1468101 (98j:47083)
  • [G-S] G. Godefroy and J. H. Shapiro, Operators with dense, invariant cyclic vector manifolds.
    J. Funct. Anal. 98 (1991), 229-269. MR 1111569 (92d:47029)
  • [G1] J. A. Goldstein,Some remarks on infinitesimal generators of analytic semigroups. Proc. Amer. Math. Soc. 22 (1969), 91-93. MR 0243384 (39:4706)
  • [G2] J. A. Goldstein, Abstract evolution equations. Trans. Amer. Math. Soc. 141 (1969), 159-185. MR 0247524 (40:789)
  • [G3] J. A. Goldstein, Semigroups of Linear Operators and Applications. Oxford University Press, New York, 1985. MR 790497 (87c:47056)
  • [GMR] J. A. Goldstein, R. M. Mininni and S. Romanelli,A new explicit formula for the solution of the Black-Merton-Scholes equation, in Infinite Dimensional Stochastic Analysis (ed. by A. Sengupta and P. Sundar), World Series Publ., 2008, 226-235. MR 2412891 (2009k:60132)
  • [GE1] K.-G. Grosse-Erdmann,Universal families and hypercyclic operators. Bull. Amer. Math. Soc. (N.S.) 36 (1999), 345-381. MR 1685272 (2000c:47001)
  • [GE2] K.-G. Grosse-Erdmann,Recent developments in hypercyclicity. RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 97 (2003), 273-286. MR 2068180 (2005c:47010)
  • [McC] C. R. MacCluer,Chaos in linear distributed systems. J. Dynam. Systems Measurement Control 114 (1992), 322-324.
  • [M-T] M. Matsui and F. Takeo, Chaotic semigroups generated by certain differential operators of order 1. SUT J. Math. 37 (2001), 51-67. MR 1849367 (2002f:47087)
  • [P-A] V. Protopopescu and Y. Y. Azmy,Topological chaos for a class of linear models. Math. Models Methods Appl. Sci. 2 (1992), 79-90. MR 1159477 (93c:58126)
  • [Tak] F. Takeo, Chaos and hypercyclicity for solution semigroups to some partial differential equations. Nonlinear Analysis63 (2005), 1943-1953.

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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

Gisèle Ruiz Goldstein
Affiliation: Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152

Jerome A. Goldstein
Affiliation: Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152

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.

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