May  2020, 13(5): 1513-1528. doi: 10.3934/dcdss.2020085

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

1. 

Department of Mathematical Sciences, University of Memphis, Memphis, 373 Dunn Hall, TN 38152-3240, USA

2. 

Department of Mathematics, University of Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy

To Angelo Favini, a great mathematician and friend

Received  February 2018 Revised  September 2018 Published  June 2019

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
$ [0, \infty) $
which can grow at infinity. We focus on the Banach spaces
$ Y_{s} = \left\{f\in C[0,\infty): \dfrac{f(x)}{1+x^{s}}\in C_0[0,\infty)\right\},\qquad s\ge 1, $
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.
Citation: Giséle Ruiz Goldstein, Jerome A. Goldstein, Rosa Maria Mininni, Silvia Romanelli. A generalized Cox-Ingersoll-Ross equation with growing initial conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1513-1528. doi: 10.3934/dcdss.2020085
References:
[1]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ, 81 (1973), 637-654.  doi: 10.1086/260062.

[2]

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.

[3]

J. C. CoxJ. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.

[4]

D. DuffieD. Filipović and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab., 13 (2003), 984-1053.  doi: 10.1214/aoap/1060202833.

[5]

R. J. Duffin, E. L. Peterson and C. Zener, Geometric Programming, John Wiley, New York, 1967.

[6]

H. EmamiradG. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation, Proc. Amer. Math. Soc., 140 (2012), 2043-2052.  doi: 10.1090/S0002-9939-2011-11069-4.

[7]

H. EmamiradG. R. Goldstein and J. A. Goldstein, Corrigendum and improvement for "Chaotic solution for the Black-Scholes equation", Proc. Amer. Math. Soc., 142 (2014), 4385-4386.  doi: 10.1090/S0002-9939-2014-12135-6.

[8]

G. R. GoldsteinJ. A. GoldsteinR. M. Mininni and S. Romanelli, The semigroup governing the generalized Cox-Ingersoll-Ross equation, Adv. Differential Equations, 21 (2016), 235-264. 

[9]

J. A. Goldstein, Cosine functions and the Feynman-Kac Formula, Quart. J. Math. Oxford, 33 (1982), 303-307.  doi: 10.1093/qmath/33.3.303.

[10]

J. A. Goldstein, Semigroups of Linear Operators and Applications, 2$^{nd}$ expanded edition, Dover Publications, 2017.

[11]

J. A. GoldsteinR. 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.

[12]

R. R. Kallman and G.-C. Rota, On the inequality $||f'||\le 4 ||f||\cdot||f''||$, in Inequalities, II (ed. O. Shisha), Academic Press, (1970), 187–192.

show all references

References:
[1]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ, 81 (1973), 637-654.  doi: 10.1086/260062.

[2]

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.

[3]

J. C. CoxJ. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.

[4]

D. DuffieD. Filipović and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab., 13 (2003), 984-1053.  doi: 10.1214/aoap/1060202833.

[5]

R. J. Duffin, E. L. Peterson and C. Zener, Geometric Programming, John Wiley, New York, 1967.

[6]

H. EmamiradG. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation, Proc. Amer. Math. Soc., 140 (2012), 2043-2052.  doi: 10.1090/S0002-9939-2011-11069-4.

[7]

H. EmamiradG. R. Goldstein and J. A. Goldstein, Corrigendum and improvement for "Chaotic solution for the Black-Scholes equation", Proc. Amer. Math. Soc., 142 (2014), 4385-4386.  doi: 10.1090/S0002-9939-2014-12135-6.

[8]

G. R. GoldsteinJ. A. GoldsteinR. M. Mininni and S. Romanelli, The semigroup governing the generalized Cox-Ingersoll-Ross equation, Adv. Differential Equations, 21 (2016), 235-264. 

[9]

J. A. Goldstein, Cosine functions and the Feynman-Kac Formula, Quart. J. Math. Oxford, 33 (1982), 303-307.  doi: 10.1093/qmath/33.3.303.

[10]

J. A. Goldstein, Semigroups of Linear Operators and Applications, 2$^{nd}$ expanded edition, Dover Publications, 2017.

[11]

J. A. GoldsteinR. 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.

[12]

R. R. Kallman and G.-C. Rota, On the inequality $||f'||\le 4 ||f||\cdot||f''||$, in Inequalities, II (ed. O. Shisha), Academic Press, (1970), 187–192.

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