The semigroup governing the generalized Cox-Ingersoll-Ross equation

Abstract

The semigroup of a generalized initial value problem including, as a particular case, the Cox-Ingersoll-Ross (CIR) equation for the price of a zero-coupon bond, is studied on spaces of continuous functions on $[0,\infty]$. The main result is the first proof of the strong continuity of the CIR semigroup. We also derive a semi-explicit representation of the semigroup and a Feynman-Kac type formula, in a generalized sense, for the unique solution of the CIR initial value problem as a useful tool for understanding additional properties of the solution itself. The Feynman-Kac type formula is the second main result of this paper.