1,871
Views
0
CrossRef citations to date
0
Altmetric
Original Articles

Multi-level Monte Carlo methods with the truncated Euler–Maruyama scheme for stochastic differential equations

, , ORCID Icon &
Pages 1715-1726 | Received 23 Nov 2016, Accepted 09 Apr 2017, Published online: 07 Jun 2017

References

  • Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate, Rev. Financ. Stud. 9 (1996), pp. 385–426. doi: 10.1093/rfs/9.2.385
  • E. Allen, Modeling with Itô Stochastic Differential Equations, Mathematical Modelling: Theory and Applications 22, Springer, Dordrecht, 2007.
  • D.F. Anderson, D.J. Higham, and Y. Sun, Multilevel Monte Carlo for stochastic differential equations with small noise, SIAM J. Numer. Anal. 54 (2016), pp. 505–529. doi: 10.1137/15M1024664
  • S. Dereich and S. Li, Multilevel Monte Carlo for Lévy-driven SDEs: Central limit theorems for adaptive Euler schemes, Ann. Appl. Probab. 26 (2016), pp. 136–185. doi: 10.1214/14-AAP1087
  • S. Dereich and S. Li, Multilevel Monte Carlo implementation for SDEs driven by truncated stable processes, in Monte Carlo and Quasi-Monte Carlo Methods, R. Cools and D. Nuyens, eds., Springer, 2016, pp. 3–27.
  • C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 3rd ed., Springer Series in Synergetics 13, Springer-Verlag, Berlin, 2004.
  • M.B. Giles, Improved multilevel monte carlo convergence using the Milstein scheme, in Monte Carlo and Quasi-Monte Carlo Methods 2006, A. Keller, S. Heinrich, and H. Niederreiter eds., Springer, Berlin, Heidelberg, 2008, pp. 343–358.
  • M.B. Giles, Multilevel monte carlo path simulation, Oper. Res. 56 (2008), pp. 607–617. doi: 10.1287/opre.1070.0496
  • M.B. Giles, Multilevel Monte Carlo methods, Acta Numer. 24 (2015), pp. 259–328. doi: 10.1017/S096249291500001X
  • A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 (2011), pp. 876–902. doi: 10.1137/10081856X
  • Q. Guo, W. Liu, X. Mao, and R. Yue, The partially truncated Euler–Maruyama method and its stability and boundedness, Appl. Numer. Math. 115 (2017), pp. 235–251. doi: 10.1016/j.apnum.2017.01.010
  • N. Halidias, Constructing positivity preserving numerical schemes for the two-factor CIR model, Monte Carlo Methods Appl. 21 (2015), pp. 313–323.
  • D.J. Higham, X. Mao, and A.M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40 (2002), pp. 1041–1063. doi: 10.1137/S0036142901389530
  • Y. Hu, Semi-implicit Euler-Maruyama scheme for stiff stochastic equations, in Stochastic Analysis and Related Topics, V (Silivri, 1994), H. Körezlioğlu, B. Øksendal, and A.S. Üstünel, eds., Progr. Probab., Vol. 38, Birkhäuser Boston, Boston, MA, 1996, pp. 183–202.
  • M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc. 236 (2015), pp. v+99.
  • M. Hutzenthaler, A. Jentzen, and P.E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab. 22 (2012), pp. 1611–1641. doi: 10.1214/11-AAP803
  • M. Hutzenthaler, A. Jentzen, and P.E. Kloeden, Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations, Ann. Appl. Probab. 23 (2013), pp. 1913–1966. doi: 10.1214/12-AAP890
  • P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York) Vol. 23, Springer-Verlag, Berlin, 1992.
  • W. Liu and X. Mao, Strong convergence of the stopped Euler–Maruyama method for nonlinear stochastic differential equations, Appl. Math. Comput. 223 (2013), pp. 389–400.
  • X. Mao, Stochastic Differential Equations and Applications, 2nd ed., Horwood Publishing Limited, Chichester, 2008.
  • X. Mao, The truncated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015), pp. 370–384. doi: 10.1016/j.cam.2015.06.002
  • X. Mao, Convergence rates of the truncated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 296 (2016), pp. 362–375. doi: 10.1016/j.cam.2015.09.035
  • G.N. Milstein and M.V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer-Verlag, Berlin, 2004.
  • Y. Niu, K. Burrage, and L. Chen, Modelling biochemical reaction systems by stochastic differential equations with reflection, J. Theoret. Biol. 396 (2016), pp. 90–104. doi: 10.1016/j.jtbi.2016.02.010
  • B. Øksendal, Stochastic Differential Equations, 6th ed., Universitext Springer-Verlag, Berlin, 2003, an introduction with applications.
  • E. Platen and D. Heath, A Benchmark Approach to Quantitative Finance, Springer Finance, Springer-Verlag, Berlin, 2006.
  • S. Sabanis, A note on tamed Euler approximations, Electron. Comm. Probab. 18 (2013), pp. 47, 10. doi: 10.1214/ECP.v18-2824
  • L. Szpruch, X. Mao, D.J. Higham, and J. Pan, Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model, BIT 51 (2011), pp. 405–425. doi: 10.1007/s10543-010-0288-y
  • M.V. Tretyakov and Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal. 51 (2013), pp. 3135–3162. doi: 10.1137/120902318
  • X. Wang and S. Gan, The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Difference Equ. Appl. 19 (2013), pp. 466–490. doi: 10.1080/10236198.2012.656617
  • C. Yue, C. Huang, and F. Jiang, Strong convergence of split-step theta methods for non-autonomous stochastic differential equations, Int. J. Comput. Math. 91 (2014), pp. 2260–2275. doi: 10.1080/00207160.2013.871541