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

Abstract

We consider the initial value problem for a nonautonomous Cox-Ingersoll-Ross equation of the type $ \begin{cases} \dfrac{\partial u}{\partial t} = \nu^2(t)\, x\, \dfrac{\partial^2 u}{\partial x^2} + (\gamma (t) + \beta (t)\, x)\, \dfrac{\partial u}{\partial x} - r(t)x\, u \\ u(0, x) = f(x), \end{cases} $ for $ x\ge 0, \; t\ge 0 $, which, under suitable conditions of financial type, models the price of a zero coupon bond. As ambient space we consider $ X_0 $, the space of all continuous complex valued functions on $ [0, \infty) $ vanishing at infinity, as well as the weighted spaces$ Y_s: = \left\{f:[0, \infty)\to \mathbb{C}:\, f \, continuous, \, \frac{f(x)}{1+x^s}\in X_0\right\}, \quad s\ge 0. $We can replace $ X_0 $ by $ Y_0 $ since $ \Vert f\Vert_{X_0} = 2\Vert f\Vert_{Y_0} $ for all $ f\in X_0 $ ($ = Y_0 $ as a set). Under suitable conditions on the time dependent coefficients we prove that this problem is well posed in $ X_0 $ and in $ Y_s, $ for $ s = 0 $ and $ s>1 $. Since the nonzero constants are very common as initial data in the Cauchy problems associated with financial models, the spaces $ Y_s $, $ s>1 $, represent a better choice for the applications.