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

• *Corresponding author: Silvia Romanelli

Dedicated to Rosa Maria (Rosella) Mininni who enthusiastically and brilliantly worked as coauthor up to the last days of her short life. Her essential contributions to our research and our friendship will forever remain in our minds and in our hearts

• 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.

Mathematics Subject Classification: Primary: 35K65, 47D06, 35Q91; Secondary: 91G20.

 Citation:

•  [1] H. Brezis, W. Rosenkrantz and B. Singer (with an Appendix by P. D. Lax), On a degenerate elliptic-parabolic equation occurring in the theory of probability, Comm. Pure Appl. Math., 24 (1971), 395-416. doi: 10.1002/cpa.3160240305. [2] J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242. [3] W. G. Faris, Product formulas for perturbation of linear propagators, J. Funct. Anal., 1 (1967), 93-108. [4] S. Fornaro and G. Metafune, Analyticity of the Cox-Ingersoll-Ross semigroup, Positivity, 24 (2020), 915-931. [5] G. Ruiz Goldstein, J. A. Goldstein, R. M. Mininni and S. Romanelli, The semigroup governing the generalized Cox-Ingersoll-Ross equation, Adv. Differ. Equ., 21 (2016), 235-264. [6] G. R. Goldstein, J. A. Goldstein, R. M. Mininni and S. Romanelli, A generalized Cox-Ingersoll-Ross equation with growing initial conditions, Discrete Cont. Dyn. Systems S, 13 (2020), 1513-1528.  doi: 10.3934/dcdss.2020085. [7] J. A. Goldstein, Semigroups of Linear Operators and Applications, 2$^{nd}$ expanded edition, Dover Publications, New York, 2017. [8] J. A. Goldstein, R. M. Mininni and S. Romanelli, Markov semigroups and groups of operators, Commun. Stoch. Anal., 1 (2007), 247-262.  doi: 10.31390/cosa.1.2.05. [9] T. Kato, Integration of the equation of evolution in a Banach space, J. Math. Soc. Japan, 5 (1953), 208-234.  doi: 10.2969/jmsj/00520208.