On a Class of Problems Related to Financial Mathematics

Abstract

Previously, we found the strongly continuous semigroups governing

$$\begin{aligned} \frac{\partial u}{\partial t}= cx^{2a}\frac{\partial ^2 u}{\partial x^2}+kx^a\frac{\partial u}{\partial x} \end{aligned}$$

for \(x,t\ge 0\) and \(a=0,1\). In this paper, we do this for a variant of the above equation where \(a=\frac{1}{2}.\) We also deal with nonautonomous versions having governing operators such as

$$\begin{aligned} L_{\alpha (t),\theta (t),r(t)} u(x) \ := \alpha (t)\ x u''(x) + \left( \frac{\alpha (t)}{2} + \theta (t)\sqrt{x}\right) u'(x) -r(t) u(x). \end{aligned}$$

Here \(\alpha , \theta ,\) and r are real-valued continuous functions in \([0,+\infty )\), \(\alpha (t)>0, \) \(\theta (t)\ge 0,\, r(t)\ge 0,\) for any \(t\ge 0.\) When \(\theta =0=r\) on \([0,\infty )\), the corresponding equation reduces to a nonautonomous version of the Cox–Ingersoll–Ross (CIR) bond equation.