December  2016, 21(10): 3359-3377. doi: 10.3934/dcdsb.2016101

Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process

1. 

Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom, United Kingdom

Received  December 2015 Revised  August 2016 Published  November 2016

We study exponential integrability properties of the Cox--Ingersoll--Ross (CIR) process and its Euler--Maruyama discretizations with various types of truncation and reflection at $0$. These properties play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a class of stochastic differential equations arising in finance. We prove that both implicit and explicit Euler--Maruyama discretizations for the CIR process preserve the exponential integrability of the exact solution for a wide range of parameters, and find lower bounds on the explosion time.
Citation: Andrei Cozma, Christoph Reisinger. Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3359-3377. doi: 10.3934/dcdsb.2016101
References:
[1]

Quantitative Finance, 13 (2013), 955-966. doi: 10.1080/14697688.2013.769688.  Google Scholar

[2]

Monte Carlo Methods and Applications, 11 (2005), 355-384. doi: 10.1515/156939605777438569.  Google Scholar

[3]

Statistics and Probability Letters, 83 (2013), 602-607. doi: 10.1016/j.spl.2012.10.034.  Google Scholar

[4]

Finance and Stochastics, 11 (2007), 29-50. doi: 10.1007/s00780-006-0011-7.  Google Scholar

[5]

ESAIM: Probability and Statistics, 12 (2008), 1-11. doi: 10.1051/ps:2007030.  Google Scholar

[6]

INRIA Research Report 5396, 2007. Google Scholar

[7]

Journal of Financial Economics, 83 (2007), 123-170. Google Scholar

[8]

Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242.  Google Scholar

[9]

A. Cozma, M. Mariapragassam and C. Reisinger, Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets,, preprint, ().   Google Scholar

[10]

Applied Stochastic Models and Data Analysis, 14 (1998), 77-84. doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2.  Google Scholar

[11]

Proceedings of the Royal Society of London A, 468 (2012), 1105-1115. doi: 10.1098/rspa.2011.0505.  Google Scholar

[12]

Operations Research, 56 (2008), 607-617. doi: 10.1287/opre.1070.0496.  Google Scholar

[13]

Springer, 2004.  Google Scholar

[14]

SIAM Journal on Financial Mathematics, 2 (2011), 255-286. doi: 10.1137/090756119.  Google Scholar

[15]

Review of Financial Studies, 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327.  Google Scholar

[16]

SIAM Journal on Numerical Analysis, 40 (2002), 1041-1063. doi: 10.1137/S0036142901389530.  Google Scholar

[17]

The Journal of Computational Finance, 8 (2005), 35-62. doi: 10.21314/JCF.2005.136.  Google Scholar

[18]

Proceedings of the Royal Society of London A, 467 (2011), 1563-1576. doi: 10.1098/rspa.2010.0348.  Google Scholar

[19]

M. Hutzenthaler, A. Jentzen and X. Wang, Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations,, preprint, ().   Google Scholar

[20]

M. Hutzenthaler, A. Jentzen and M. Noll, Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundaries,, preprint, ().   Google Scholar

[21]

Mem. Amer. Math. Soc., 236 (2015), v+99 pp. doi: 10.1090/memo/1112.  Google Scholar

[22]

$2^{nd}$ edition, Springer, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[23]

Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[24]

International Journal of Differential Equations, (2011), 13 pages.  Google Scholar

[25]

Quantitative Finance, 10 (2010), 177-194. doi: 10.1080/14697680802392496.  Google Scholar

[26]

Numerische Mathematik, 128 (2014), 103-136. doi: 10.1007/s00211-014-0606-4.  Google Scholar

[27]

International Journal of Theoretical and Applied Finance, 17 (2014), 1450045, 30 pp. doi: 10.1142/S0219024914500459.  Google Scholar

show all references

References:
[1]

Quantitative Finance, 13 (2013), 955-966. doi: 10.1080/14697688.2013.769688.  Google Scholar

[2]

Monte Carlo Methods and Applications, 11 (2005), 355-384. doi: 10.1515/156939605777438569.  Google Scholar

[3]

Statistics and Probability Letters, 83 (2013), 602-607. doi: 10.1016/j.spl.2012.10.034.  Google Scholar

[4]

Finance and Stochastics, 11 (2007), 29-50. doi: 10.1007/s00780-006-0011-7.  Google Scholar

[5]

ESAIM: Probability and Statistics, 12 (2008), 1-11. doi: 10.1051/ps:2007030.  Google Scholar

[6]

INRIA Research Report 5396, 2007. Google Scholar

[7]

Journal of Financial Economics, 83 (2007), 123-170. Google Scholar

[8]

Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242.  Google Scholar

[9]

A. Cozma, M. Mariapragassam and C. Reisinger, Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets,, preprint, ().   Google Scholar

[10]

Applied Stochastic Models and Data Analysis, 14 (1998), 77-84. doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2.  Google Scholar

[11]

Proceedings of the Royal Society of London A, 468 (2012), 1105-1115. doi: 10.1098/rspa.2011.0505.  Google Scholar

[12]

Operations Research, 56 (2008), 607-617. doi: 10.1287/opre.1070.0496.  Google Scholar

[13]

Springer, 2004.  Google Scholar

[14]

SIAM Journal on Financial Mathematics, 2 (2011), 255-286. doi: 10.1137/090756119.  Google Scholar

[15]

Review of Financial Studies, 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327.  Google Scholar

[16]

SIAM Journal on Numerical Analysis, 40 (2002), 1041-1063. doi: 10.1137/S0036142901389530.  Google Scholar

[17]

The Journal of Computational Finance, 8 (2005), 35-62. doi: 10.21314/JCF.2005.136.  Google Scholar

[18]

Proceedings of the Royal Society of London A, 467 (2011), 1563-1576. doi: 10.1098/rspa.2010.0348.  Google Scholar

[19]

M. Hutzenthaler, A. Jentzen and X. Wang, Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations,, preprint, ().   Google Scholar

[20]

M. Hutzenthaler, A. Jentzen and M. Noll, Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundaries,, preprint, ().   Google Scholar

[21]

Mem. Amer. Math. Soc., 236 (2015), v+99 pp. doi: 10.1090/memo/1112.  Google Scholar

[22]

$2^{nd}$ edition, Springer, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[23]

Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[24]

International Journal of Differential Equations, (2011), 13 pages.  Google Scholar

[25]

Quantitative Finance, 10 (2010), 177-194. doi: 10.1080/14697680802392496.  Google Scholar

[26]

Numerische Mathematik, 128 (2014), 103-136. doi: 10.1007/s00211-014-0606-4.  Google Scholar

[27]

International Journal of Theoretical and Applied Finance, 17 (2014), 1450045, 30 pp. doi: 10.1142/S0219024914500459.  Google Scholar

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