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Chaotic solution for the Black-Scholes equation. (English) Zbl 1238.47051

Proc. Am. Math. Soc. 140, No. 6, 2043-2052 (2012); corrigendum ibid. 142, No. 12, 4385-4386 (2014).
Summary: 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 \[ Y^{s,\tau }:=\left\{u\in C((0,\infty )):\lim_{x\rightarrow \infty} \frac{u(x)}{1+x^{s}}=0, \;\lim _{x\rightarrow 0}\frac{u(x)}{1+x^{-\tau }} =0\right\} \] with norm \( \left \| u\right \|_{Y^{s,\tau}}=\sup_{x>0}\left | \frac {u(x)}{(1+x^{s})(1+x^{-\tau })}\right | <\infty\), 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 [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, mathematical programming, economics
47D06 One-parameter semigroups and linear evolution equations
91G80 Financial applications of other theories
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
47A16 Cyclic vectors, hypercyclic and chaotic operators

Citations:

Zbl 0732.47016
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Full Text: DOI

References:

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