On a Class of Problems Related to Financial Mathematics

Previously, we found the strongly continuous semigroups governing \(\begin{aligned} \frac{\partial u}{\partial t}= cx^{2a}\frac{\partial ^2 u}{\partial x^2}+kx^a\frac{\partial u}{\partial x} \end{aligned}\) $$\begin{array}{c}\hfill \frac{\partial u}{\partial t}=c{x}^{2a}\frac{{\partial }^{2}u}{\partial x^2}+k{x}^a\frac{\partial u}{\partial x}\end{array}$$ for \(x,t\ge 0\) $x,t\ge 0$ and \(a=0,1\) $a=0,1$ . In this paper, we do this for a variant of the above equation where \(a=\frac{1}{2}.\) $a=\frac{1}{2}.$ We also deal with nonautonomous versions having governing operators such as \(\begin{aligned} L_{\alpha (t),\theta (t),r(t)} u(x) \ := \alpha (t)\ x u''(x) + \left( \frac{\alpha (t)}{2} + \theta (t)\sqrt{x}\right) u'(x) -r(t) u(x). \end{aligned}\) $$\begin{array}{c}\hfill L_{\alpha (t),\theta (t),r(t)}u(x):=\alpha (t)x{u}^{\prime \prime }(x)+\left(\frac{\alpha (t)}{2},+,\theta ,(t),\sqrt{x}\right)u^{\prime }(x)-r(t)u(x).\end{array}$$ Here \(\alpha , \theta ,\) $\alpha ,\theta ,$ and r are real-valued continuous functions in \([0,+\infty )\) $[0,+\infty )$ , \(\alpha (t)>0, \) $\alpha (t)>0,$ \(\theta (t)\ge 0,\, r(t)\ge 0,\) $\theta (t)\ge 0,r(t)\ge 0,$ for any \(t\ge 0.\) $t\ge 0.$ When \(\theta =0=r\) $\theta =0=r$ on \([0,\infty )\) $[0,\infty )$ , the corresponding equation reduces to a nonautonomous version of the Cox–Ingersoll–Ross (CIR) bond equation.