In Emamirad et al. (in: Semigroups of Operators - Theory and Applications, Springer, 2014; and Proc. Amer. Math. Soc. 140:2043–2052, 2012), the Black–Scholes semigroup was studied in various Banach spaces of continuous functions regarding chaotic behavior and the null volatility limit. Here we let the volatility and the interest rates be continuous functions on the half line $$[0,\infty )$$ [ 0 , ∞ ) , and we consider the generalized Black–Scholes equation as a linear nonautonomous abstract Cauchy problem. It is shown that this problem is wellposed and an explicit formula for the corresponding strongly continuous evolution family is given. Furthermore, a hypercyclicity criterion for strongly continuous evolution families on separable Banach spaces is proved and applied to the Black–Scholes evolution family. Finally, the chaotic behavior is defined for strongly continuous evolution families and it is shown that the Black–Scholes evolution family is chaotic when the volatility and the interest rates are periodic functions.

中文翻译：

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具有时间相关系数的混沌 Black-Scholes 方程

在 Emamirad 等人中。(in: Semigroups of Operators - Theory and Applications, Springer, 2014; and Proc. Amer. Math. Soc. 140:2043–2052, 2012)，Black-Scholes 半群在连续函数的各种 Banach 空间中进行了关于混沌行为的研究和零波动率限制。在这里，我们让波动率和利率是半线 $$[0,\infty )$$ [ 0 , ∞ ) 上的连续函数，我们将广义 Black-Scholes 方程视为线性非自治抽象柯西问题。结果表明，该问题是适定的，并给出了对应的强连续演化族的显式公式。此外，证明了可分离 Banach 空间上强连续演化族的超循环性准则，并将其应用于 Black-Scholes 演化族。最后，