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Convergence, non-negativity and stability of a new Milstein scheme with applications to finance
1. | Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, United Kingdom |
2. | Department of Statistics and Modelling Science, University of Strathclyde, Glasgow, G1 1XH, Scotland |
3. | School of Mathematics, The University of Edinburgh, Edinburgh, EH9 3JZ, United Kingdom |
References:
[1] |
Assyr Abdulle and Adrian Blumenthal, Stabilized multilevel Monte Carlo method for stiff stochastic differential equations, EPFL-ARTICLE-183502, (2013).
doi: 10.1016/j.jcp.2013.05.039. |
[2] |
D. H. Ahn and B. Gao, A parametric nonlinear model of term structure dynamics, Review of Financial Studies, 12 (1999), 721.
doi: 10.1093/rfs/12.4.721. |
[3] |
Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate, Review of Financial Studies, 9 (1996), 385-426. |
[4] |
J. A. D. Appleby, M. Guzowska, C. Kelly and A. Rodkina, Preserving positivity in solutions of discretised stochastic differential equations, Applied Mathematics and Computation, 217 (2010), 763-774.
doi: 10.1016/j.amc.2010.06.015. |
[5] |
E. Buckwar and T. Sickenberger, A comparative linear mean-square stability analysis of Maruyama-and Milstein-type methods, Mathematics and Computers in Simulation, 81 (2011), 1110-1127.
doi: 10.1016/j.matcom.2010.09.015. |
[6] |
T. C. Gard, "Introduction to Stochastic Differential Equations," Marcel Dekker, New York, 1988. |
[7] |
M. Giles, Improved multilevel Monte Carlo convergence using the Milstein scheme, Monte Carlo and Quasi-Monte Carlo Methods, 2006 (2008), 343-358.
doi: 10.1007/978-3-540-74496-2_20. |
[8] |
M. B. Giles, Multilevel Monte Carlo path simulation, Operations Research-Baltimore, 56 (2008), 607-617.
doi: 10.1287/opre.1070.0496. |
[9] |
J. Goard and M. Mazur, "Stochastic Volatility Models and the Pricing of Vix Options," Mathematical Finance, 2011.
doi: 10.1111/j.1467-9965.2011.00506.x. |
[10] |
I. Gyöngy, A note on Euler's approximations, Potential Analysis, 8 (1998), 205-216.
doi: 10.1023/A:1008605221617. |
[11] |
S. L. Heston, "A Simple New Formula for Options with Stochastic Volatility," Course Notes of Washington University in St. Louis, Missouri, 1997. |
[12] |
D. J. Higham, A-stability and stochastic mean-square stability, BIT Numerical Mathematics, 40 (2000), 404-409.
doi: 10.1023/A:1022355410570. |
[13] |
D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM Journal on Numerical Analysis, 38 (2000), 753-769.
doi: 10.1137/S003614299834736X. |
[14] |
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 A, 467 (2011), 1563-1576. |
[15] |
M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, The Annals of Applied Probability, 22 (2012), 1611-1641.
doi: 10.1214/11-AAP803. |
[16] |
A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients, Numerische Mathematik, 112 (2009), 41-64.
doi: 10.1007/s00211-008-0200-8. |
[17] |
C. Kahl, M. Gunther and T. Rosberg, Structure preserving stochastic integration schemes in interest rate derivative modeling, Applied Numerical Mathematics, 58 (2008), 284-295.
doi: 10.1016/j.apnum.2006.11.013. |
[18] |
C. Kahl and H. Schurz, Balanced Milstein methods for ordinary SDEs, Monte Carlo Methods and Applications, 12 (2006), 143-170.
doi: 10.1515/156939606777488842. |
[19] |
I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Springer, 1991.
doi: 10.1007/978-1-4612-0949-2. |
[20] |
P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer, 1992. |
[21] |
A. L. Lewis, "Option Valuation Under Stochastic Volatility," Finance Press, 2000. |
[22] |
R. S. Liptser and A. N. Shiryayev, "Theory of Martingales," Kluwer Academic Publishers, 1989.
doi: 10.1007/978-94-009-2438-3. |
[23] |
R. Lord, R. Koekkoek and D. J. C. Van Dijk, A comparison of biased simulation schemes for stochastic volatility models, Quantitative Finance, 10 (2010), 177-194.
doi: 10.1080/14697680802392496. |
[24] |
X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingales," Longman Scientific & Technical, 1991. |
[25] |
X. Mao, "Stochastic Differential Equations and Their Applications," Horwood Pub Ltd, 1997. |
[26] |
X. Mao and L. Szpruch, Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients, Stochastics, 85 (2012), 144-177.
doi: 10.1080/17442508.2011.651213. |
[27] |
X. Mao and L. Szpruch, Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Comput. Appl. Math., 238 (2013), 14-28.
doi: 10.1016/j.cam.2012.08.015. |
[28] |
G. N. Milstein, E. Platen and H. Schurz, Balanced implicit methods for stiff stochastic systems, SIAM Journal on Numerical Analysis, 35 (1998), 1010-1019.
doi: 10.1137/S0036142994273525. |
[29] |
G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics. Scientific Computation," Springer-Verlag, Berlin, 2004. |
[30] |
Tetsuya Misawa, A lie algebraic approach to numerical integration of stochastic differential equations, SIAM Journal on Scientific Computing, 23 (2001), 866-890.
doi: 10.1137/S106482750037024X. |
[31] |
A. Neuenkirch and L. Szpruch, First order strong approximations of scalar sdes with values in a domain,, Preprint. , ().
|
[32] |
H. Schurz, Convergence and stability of balanced implicit methods for systems of SDEs, Int. J. Numer. Anal. Model, 2 (2005), 197-220. |
[33] |
Y. Shen, Q. Luo and X. Mao, The improved LaSalle-type theorems for stochastic functional differential equations, Journal of Mathematical Analysis and Applications, 318 (2006), 134-154.
doi: 10.1016/j.jmaa.2005.05.026. |
[34] |
A. N. Shiryaev, "Probability," Springer-Verlag, New York, 1996. |
[35] |
L. Szpruch, X. Mao, D. J. Higham and J. Pan, Numerical simulation of a strongly nonlinear Ait-Sahalia type interest rate model, BIT Numerical Mathematics, 51 (2011), 405-425.
doi: 10.1007/s10543-010-0288-y. |
[36] |
X. Wang, S. Gan and D. Wang, A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise, BIT, 52 (2012), 741-772.
doi: 10.1007/s10543-012-0370-8. |
[37] |
E. Zeidler, "Nonlinear Functional Analysis and Its Applications," Springer Verlag, 1990. |
show all references
References:
[1] |
Assyr Abdulle and Adrian Blumenthal, Stabilized multilevel Monte Carlo method for stiff stochastic differential equations, EPFL-ARTICLE-183502, (2013).
doi: 10.1016/j.jcp.2013.05.039. |
[2] |
D. H. Ahn and B. Gao, A parametric nonlinear model of term structure dynamics, Review of Financial Studies, 12 (1999), 721.
doi: 10.1093/rfs/12.4.721. |
[3] |
Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate, Review of Financial Studies, 9 (1996), 385-426. |
[4] |
J. A. D. Appleby, M. Guzowska, C. Kelly and A. Rodkina, Preserving positivity in solutions of discretised stochastic differential equations, Applied Mathematics and Computation, 217 (2010), 763-774.
doi: 10.1016/j.amc.2010.06.015. |
[5] |
E. Buckwar and T. Sickenberger, A comparative linear mean-square stability analysis of Maruyama-and Milstein-type methods, Mathematics and Computers in Simulation, 81 (2011), 1110-1127.
doi: 10.1016/j.matcom.2010.09.015. |
[6] |
T. C. Gard, "Introduction to Stochastic Differential Equations," Marcel Dekker, New York, 1988. |
[7] |
M. Giles, Improved multilevel Monte Carlo convergence using the Milstein scheme, Monte Carlo and Quasi-Monte Carlo Methods, 2006 (2008), 343-358.
doi: 10.1007/978-3-540-74496-2_20. |
[8] |
M. B. Giles, Multilevel Monte Carlo path simulation, Operations Research-Baltimore, 56 (2008), 607-617.
doi: 10.1287/opre.1070.0496. |
[9] |
J. Goard and M. Mazur, "Stochastic Volatility Models and the Pricing of Vix Options," Mathematical Finance, 2011.
doi: 10.1111/j.1467-9965.2011.00506.x. |
[10] |
I. Gyöngy, A note on Euler's approximations, Potential Analysis, 8 (1998), 205-216.
doi: 10.1023/A:1008605221617. |
[11] |
S. L. Heston, "A Simple New Formula for Options with Stochastic Volatility," Course Notes of Washington University in St. Louis, Missouri, 1997. |
[12] |
D. J. Higham, A-stability and stochastic mean-square stability, BIT Numerical Mathematics, 40 (2000), 404-409.
doi: 10.1023/A:1022355410570. |
[13] |
D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM Journal on Numerical Analysis, 38 (2000), 753-769.
doi: 10.1137/S003614299834736X. |
[14] |
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 A, 467 (2011), 1563-1576. |
[15] |
M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, The Annals of Applied Probability, 22 (2012), 1611-1641.
doi: 10.1214/11-AAP803. |
[16] |
A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients, Numerische Mathematik, 112 (2009), 41-64.
doi: 10.1007/s00211-008-0200-8. |
[17] |
C. Kahl, M. Gunther and T. Rosberg, Structure preserving stochastic integration schemes in interest rate derivative modeling, Applied Numerical Mathematics, 58 (2008), 284-295.
doi: 10.1016/j.apnum.2006.11.013. |
[18] |
C. Kahl and H. Schurz, Balanced Milstein methods for ordinary SDEs, Monte Carlo Methods and Applications, 12 (2006), 143-170.
doi: 10.1515/156939606777488842. |
[19] |
I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Springer, 1991.
doi: 10.1007/978-1-4612-0949-2. |
[20] |
P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer, 1992. |
[21] |
A. L. Lewis, "Option Valuation Under Stochastic Volatility," Finance Press, 2000. |
[22] |
R. S. Liptser and A. N. Shiryayev, "Theory of Martingales," Kluwer Academic Publishers, 1989.
doi: 10.1007/978-94-009-2438-3. |
[23] |
R. Lord, R. Koekkoek and D. J. C. Van Dijk, A comparison of biased simulation schemes for stochastic volatility models, Quantitative Finance, 10 (2010), 177-194.
doi: 10.1080/14697680802392496. |
[24] |
X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingales," Longman Scientific & Technical, 1991. |
[25] |
X. Mao, "Stochastic Differential Equations and Their Applications," Horwood Pub Ltd, 1997. |
[26] |
X. Mao and L. Szpruch, Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients, Stochastics, 85 (2012), 144-177.
doi: 10.1080/17442508.2011.651213. |
[27] |
X. Mao and L. Szpruch, Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Comput. Appl. Math., 238 (2013), 14-28.
doi: 10.1016/j.cam.2012.08.015. |
[28] |
G. N. Milstein, E. Platen and H. Schurz, Balanced implicit methods for stiff stochastic systems, SIAM Journal on Numerical Analysis, 35 (1998), 1010-1019.
doi: 10.1137/S0036142994273525. |
[29] |
G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics. Scientific Computation," Springer-Verlag, Berlin, 2004. |
[30] |
Tetsuya Misawa, A lie algebraic approach to numerical integration of stochastic differential equations, SIAM Journal on Scientific Computing, 23 (2001), 866-890.
doi: 10.1137/S106482750037024X. |
[31] |
A. Neuenkirch and L. Szpruch, First order strong approximations of scalar sdes with values in a domain,, Preprint. , ().
|
[32] |
H. Schurz, Convergence and stability of balanced implicit methods for systems of SDEs, Int. J. Numer. Anal. Model, 2 (2005), 197-220. |
[33] |
Y. Shen, Q. Luo and X. Mao, The improved LaSalle-type theorems for stochastic functional differential equations, Journal of Mathematical Analysis and Applications, 318 (2006), 134-154.
doi: 10.1016/j.jmaa.2005.05.026. |
[34] |
A. N. Shiryaev, "Probability," Springer-Verlag, New York, 1996. |
[35] |
L. Szpruch, X. Mao, D. J. Higham and J. Pan, Numerical simulation of a strongly nonlinear Ait-Sahalia type interest rate model, BIT Numerical Mathematics, 51 (2011), 405-425.
doi: 10.1007/s10543-010-0288-y. |
[36] |
X. Wang, S. Gan and D. Wang, A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise, BIT, 52 (2012), 741-772.
doi: 10.1007/s10543-012-0370-8. |
[37] |
E. Zeidler, "Nonlinear Functional Analysis and Its Applications," Springer Verlag, 1990. |
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