A generalized Cox-Ingersoll-Ross equation with growing initial conditions

Abstract

In this paper we solve the problem of the existence and strong continuity of the semigroup associated with the initial value problem for a generalized Cox-Ingersoll-Ross equation for the price of a zero-coupon bond (see [ 8 ]), on spaces of continuous functions on \begin{document}$ [0, \infty) $\end{document} which can grow at infinity. We focus on the Banach spaces \begin{document}$ Y_{s} = \left\{f\in C[0,\infty): \dfrac{f(x)}{1+x^{s}}\in C_0[0,\infty)\right\},\qquad s\ge 1, $\end{document} which contain the nonzero constants very common as initial data in the Cauchy problems coming from financial models. In addition, a Feynman-Kac type formula is given.