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

Summary: The Black-Scholes semigroup is studied on spaces of continuous functions on (0,) which may grow at both 0 and at , which is important since the standard initial value is an unbounded function. We prove that in the Banach spaces

Y s,τ :=uC((0,)):lim x u(x) 1+x s =0,lim x0 u(x) 1+x -τ =0

with norm u Y s,τ =sup x>0 u(x) (1+x s )(1+x -τ )<, the Black-Scholes semigroup is strongly continuous and chaotic for s>1, τ0 with sν>1, where 2ν 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:
47N10Applications of operator theory in optimization, convex analysis, programming, economics
47D06One-parameter semigroups and linear evolution equations
91G80Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
35Q91PDEs in connection with game theory, economics, social and behavioral sciences
47A16Cyclic vectors, hypercyclic and chaotic operators