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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[1]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[2]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[3]

Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263

[4]

Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020386

[5]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[6]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

[7]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[8]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460

[9]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[10]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[11]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[12]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[13]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[14]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[15]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[16]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[17]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[18]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[19]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[20]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (129)
  • HTML views (419)
  • Cited by (1)

[Back to Top]