The semigroup governing the generalized Cox-Ingersoll-Ross equation. (English) Zbl 1341.47051

Goldstein, Gisèle Ruiz; Goldstein, Jerome A.; Mininni, Rosa Maria; Romanelli, Silvia

Adv. Differ. Equ. 21, No. 3-4, 235-264 (2016).

Summary: 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.

MSC:

47D06	One-parameter semigroups and linear evolution equations
35K15	Initial value problems for second-order parabolic equations
35C15	Integral representations of solutions to PDEs
91G20	Derivative securities (option pricing, hedging, etc.)