American Institute of Mathematical Sciences

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:
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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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