# 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 and Continuous Dynamical Systems - B, 2016, 21 (10) : 3359-3377. doi: 10.3934/dcdsb.2016101
##### References:
 [1] R. Ahlip and M. Rutkowski, Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates, Quantitative Finance, 13 (2013), 955-966. doi: 10.1080/14697688.2013.769688. [2] A. Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes, Monte Carlo Methods and Applications, 11 (2005), 355-384. doi: 10.1515/156939605777438569. [3] A. Alfonsi, Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process, Statistics and Probability Letters, 83 (2013), 602-607. doi: 10.1016/j.spl.2012.10.034. [4] L. Andersen and V. Piterbarg, Moment explosions in stochastic volatility models, Finance and Stochastics, 11 (2007), 29-50. doi: 10.1007/s00780-006-0011-7. [5] A. Berkaoui, M. Bossy and A. Diop, Euler scheme for SDEs with non-Lipschitz diffusion coefficient: Strong convergence, ESAIM: Probability and Statistics, 12 (2008), 1-11. doi: 10.1051/ps:2007030. [6] M. Bossy and A. Diop, An Efficient Discretization Scheme for one Dimensional SDEs with a Diffusion Coefficient Function of the Form $|x|^{\alpha}$, $\alpha \in [1/2,1)$, INRIA Research Report 5396, 2007. [7] P. Cheridito, D. Filipović and R. L. Kimmel, Market price of risk specifications for affine models: Theory and evidence, Journal of Financial Economics, 83 (2007), 123-170. [8] J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242. [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, (). [10] G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term, 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. [11] S. Dereich, A. Neuenkirch and L. Szpruch, An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process, Proceedings of the Royal Society of London A, 468 (2012), 1105-1115. doi: 10.1098/rspa.2011.0505. [12] M. Giles, Multilevel Monte Carlo path simulation, Operations Research, 56 (2008), 607-617. doi: 10.1287/opre.1070.0496. [13] P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2004. [14] L. A. Grzelak and C. W. Oosterlee, On the Heston model with stochastic interest rates, SIAM Journal on Financial Mathematics, 2 (2011), 255-286. doi: 10.1137/090756119. [15] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327. [16] D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis, 40 (2002), 1041-1063. doi: 10.1137/S0036142901389530. [17] D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process, The Journal of Computational Finance, 8 (2005), 35-62. doi: 10.21314/JCF.2005.136. [18] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proceedings of the Royal Society of London A, 467 (2011), 1563-1576. doi: 10.1098/rspa.2010.0348. [19] M. Hutzenthaler, A. Jentzen and X. Wang, Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations,, preprint, (). [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, (). [21] M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc., 236 (2015), v+99 pp. doi: 10.1090/memo/1112. [22] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, $2^{nd}$ edition, Springer, 1991. doi: 10.1007/978-1-4612-0949-2. [23] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. [24] S. R. Liberty and L. Mou, Existence of solutions of a Riccati differential system from a general cumulant control problem, International Journal of Differential Equations, (2011), 13 pages. [25] R. Lord, R. Koekkoek and D. Van Dijk, A comparison of biased simulation schemes for stochastic volatility models, Quantitative Finance, 10 (2010), 177-194. doi: 10.1080/14697680802392496. [26] A. Neuenkirch and L. Szpruch, First order strong approximations of scalar SDEs defined in a domain, Numerische Mathematik, 128 (2014), 103-136. doi: 10.1007/s00211-014-0606-4. [27] A. Van der Stoep, L. A. Grzelak and C. W. Oosterlee, The Heston stochastic-local volatility model: Efficient Monte Carlo simulation, International Journal of Theoretical and Applied Finance, 17 (2014), 1450045, 30 pp. doi: 10.1142/S0219024914500459.

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##### References:
 [1] R. Ahlip and M. Rutkowski, Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates, Quantitative Finance, 13 (2013), 955-966. doi: 10.1080/14697688.2013.769688. [2] A. Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes, Monte Carlo Methods and Applications, 11 (2005), 355-384. doi: 10.1515/156939605777438569. [3] A. Alfonsi, Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process, Statistics and Probability Letters, 83 (2013), 602-607. doi: 10.1016/j.spl.2012.10.034. [4] L. Andersen and V. Piterbarg, Moment explosions in stochastic volatility models, Finance and Stochastics, 11 (2007), 29-50. doi: 10.1007/s00780-006-0011-7. [5] A. Berkaoui, M. Bossy and A. Diop, Euler scheme for SDEs with non-Lipschitz diffusion coefficient: Strong convergence, ESAIM: Probability and Statistics, 12 (2008), 1-11. doi: 10.1051/ps:2007030. [6] M. Bossy and A. Diop, An Efficient Discretization Scheme for one Dimensional SDEs with a Diffusion Coefficient Function of the Form $|x|^{\alpha}$, $\alpha \in [1/2,1)$, INRIA Research Report 5396, 2007. [7] P. Cheridito, D. Filipović and R. L. Kimmel, Market price of risk specifications for affine models: Theory and evidence, Journal of Financial Economics, 83 (2007), 123-170. [8] J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242. [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, (). [10] G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term, 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. [11] S. Dereich, A. Neuenkirch and L. Szpruch, An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process, Proceedings of the Royal Society of London A, 468 (2012), 1105-1115. doi: 10.1098/rspa.2011.0505. [12] M. Giles, Multilevel Monte Carlo path simulation, Operations Research, 56 (2008), 607-617. doi: 10.1287/opre.1070.0496. [13] P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2004. [14] L. A. Grzelak and C. W. Oosterlee, On the Heston model with stochastic interest rates, SIAM Journal on Financial Mathematics, 2 (2011), 255-286. doi: 10.1137/090756119. [15] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327. [16] D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis, 40 (2002), 1041-1063. doi: 10.1137/S0036142901389530. [17] D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process, The Journal of Computational Finance, 8 (2005), 35-62. doi: 10.21314/JCF.2005.136. [18] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proceedings of the Royal Society of London A, 467 (2011), 1563-1576. doi: 10.1098/rspa.2010.0348. [19] M. Hutzenthaler, A. Jentzen and X. Wang, Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations,, preprint, (). [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, (). [21] M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc., 236 (2015), v+99 pp. doi: 10.1090/memo/1112. [22] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, $2^{nd}$ edition, Springer, 1991. doi: 10.1007/978-1-4612-0949-2. [23] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. [24] S. R. Liberty and L. Mou, Existence of solutions of a Riccati differential system from a general cumulant control problem, International Journal of Differential Equations, (2011), 13 pages. [25] R. Lord, R. Koekkoek and D. Van Dijk, A comparison of biased simulation schemes for stochastic volatility models, Quantitative Finance, 10 (2010), 177-194. doi: 10.1080/14697680802392496. [26] A. Neuenkirch and L. Szpruch, First order strong approximations of scalar SDEs defined in a domain, Numerische Mathematik, 128 (2014), 103-136. doi: 10.1007/s00211-014-0606-4. [27] A. Van der Stoep, L. A. Grzelak and C. W. Oosterlee, The Heston stochastic-local volatility model: Efficient Monte Carlo simulation, International Journal of Theoretical and Applied Finance, 17 (2014), 1450045, 30 pp. doi: 10.1142/S0219024914500459.
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