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Optimal control of a perturbed sweeping process via discrete approximations
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 |
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. |
show all references
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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