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August  2019, 24(8): 3475-3502. doi: 10.3934/dcdsb.2018253

A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients

Raphael Kruse and  Yue Wu

Technische Universität Berlin, Institut für Mathematik, Secr. MA 5-3, Straße des 17. Juni 136, DE-10623 Berlin, Germany

Received  September 2017 Revised  February 2018 Published  August 2019 Early access  August 2018

Fund Project: The authors are supported by the German Research Foundation through FOR 2402.

In this paper a drift-randomized Milstein method is introduced for the numerical solution of non-autonomous stochastic differential equations with non-differentiable drift coefficient functions. Compared to standard Milstein-type methods we obtain higher order convergence rates in the $ L^p(Ω) $ and almost sure sense. An important ingredient in the error analysis are randomized quadrature rules for Hölder continuous stochastic processes. By this we avoid the use of standard arguments based on the Itō-Taylor expansion which are typically applied in error estimates of the classical Milstein method but require additional smoothness of the drift and diffusion coefficient functions. We also discuss the optimality of our convergence rates. Finally, the question of implementation is addressed in a numerical experiment.

Keywords: Milstein method, stochastic differential equations, strong convergence, Monte Carlo methods, randomized quadrature rules.
Mathematics Subject Classification: Primary: 65C30; Secondary: 65C05, 65L20, 60H10.
Citation: Raphael Kruse, Yue Wu. A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3475-3502. doi: 10.3934/dcdsb.2018253
References:
[1]

W.-J. Beyn and R. Kruse, Two-sided error estimates for the stochastic theta method, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 389-407, URL http://dx.doi.org/10.3934/dcdsb.2010.14.389. doi: 10.3934/dcdsb.2010.14.389.

[2]

D. L. Burkholder, Martingale transforms, Ann. Math. Statist., 37 (1966), 1494-1504.  doi: 10.1214/aoms/1177699141.

[3]

J. M. C. Clark and R. J. Cameron, The maximum rate of convergence of discrete approximations for stochastic differential equations, in Stochastic Differential Systems (Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978), vol. 25 of Lecture Notes in Control and Information Sci., Springer, Berlin, 1980,162-171.

[4]

T. Daun, On the randomized solution of initial value problems, J. Complexity, 27 (2011), 300-311, URL http://dx.doi.org/10.1016/j.jco.2010.07.002. doi: 10.1016/j.jco.2010.07.002.

[5]

E. Emmrich, Discrete versions of Gronwall's lemma and their application to the numerical analysis of parabolic problems, TU Berlin, FB Mathematik, Preprint.

[6]

J. G. Gaines and T. J. Lyons, Random generation of stochastic area integrals, SIAM J. Appl. Math., 54 (1994), 1132-1146, URL http://dx.doi.org/10.1137/S0036139992235706. doi: 10.1137/S0036139992235706.

[7]

M. B. Giles and L. Szpruch, Antithetic multilevel Monte Carlo estimation for multidimensional SDEs without Lévy area simulation, Ann. Appl. Probab., 24 (2014), 1585-1620, URL http://dx.doi.org/10.1214/13-AAP957. doi: 10.1214/13-AAP957.

[8]

I. Gyöngy, A note on Euler's approximations, Potential Anal. , 8 (1998), 205-216, URL http://dx.doi.org/10.1023/A:1008605221617.

[9]

S. Haber, A modified Monte-Carlo quadrature, Math. Comp., 20 (1966), 361-368.  doi: 10.1090/S0025-5718-1966-0210285-0.

[10]

S. Haber, A modified Monte-Carlo quadrature. Ⅱ, Math. Comp., 21 (1967), 388-397.  doi: 10.1090/S0025-5718-1967-0234606-9.

[11]

S. Heinrich and B. Milla, The randomized complexity of initial value problems, J. Complexity, 24 (2008), 77-88.  doi: 10.1016/j.jco.2007.09.002.

[12]

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, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 1563-1576, URL http://dx.doi.org/10.1098/rspa.2010.0348. doi: 10.1098/rspa.2010.0348.

[13]

A. Jentzen and A. Neuenkirch, A random Euler scheme for Carathéodory differential equations, J. Comput. Appl. Math., 224 (2009), 346-359, URL http://dx.doi.org/10.1016/j.cam.2008.05.060. doi: 10.1016/j.cam.2008.05.060.

[14]

B. Kacewicz, Optimal solution of ordinary differential equations, J. Complexity, 3 (1987), 451-465, URL http://dx.doi.org/10.1016/0885-064X(87)90011-2. doi: 10.1016/0885-064X(87)90011-2.

[15]

B. Kacewicz, Asymptotic setting (revisited): Analysis of a boundary-value problem and a relation to a classical approximation result, J. Complexity, 20 (2004), 796-806, URL https://doi.org/10.1016/j.jco.2003.08.006. doi: 10.1016/j.jco.2003.08.006.

[16]

B. Kacewicz, Almost optimal solution of initial-value problems by randomized and quantum algorithms, J. Complexity, 22 (2006), 676-690, URL http://dx.doi.org/10.1016/j.jco.2006.03.001. doi: 10.1016/j.jco.2006.03.001.

[17]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, vol. 113 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, New York, 1991, URL https://doi.org/10.1007/978-1-4612-0949-2. doi: 10.1007/978-1-4612-0949-2.

[18]

P. E. Kloeden and A. Neuenkirch, The pathwise convergence of approximation schemes for stochastic differential equations, LMS J. Comput. Math., 10 (2007), 235-253, URL http://dx.doi.org/10.1112/S1461157000001388. doi: 10.1112/S1461157000001388.

[19]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, 3rd edition, Springer, Berlin, 1999. doi: 10.1007/978-3-662-12616-5.

[20]

R. Kruse, Characterization of bistability for stochastic multistep methods, BIT, 52 (2012), 109-140, URL http://dx.doi.org/10.1007/s10543-011-0341-5. doi: 10.1007/s10543-011-0341-5.

[21]

R. Kruse, Consistency and stability of a Milstein-Galerkin finite element scheme for semilinear SPDE, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014), 471-516, URL http://dx.doi.org/10.1007/s40072-014-0037-3. doi: 10.1007/s40072-014-0037-3.

[22]

R. Kruse and Y. Wu, Error analysis of randomized Runge-Kutta methods for differential equations with time-irregular coefficients, Comput. Methods Appl. Math., 17 (2017), 479-498.  doi: 10.1515/cmam-2016-0048.

[23]

X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood Publishing Limited, Chichester, 2008, URL http://dx.doi.org/10.1533/9780857099402. doi: 10.1533/9780857099402.

[24]

G. N. Milstein, Approximate integration of stochastic differential equations, Teor. Verojatnost. i Primenen., 19 (1974), 583-588, In Russian.

[25]

G. N. Milstein, Approximate integration of stochastic differential equations, Theory Probab. Appl., 19 (1975), 557-562, Translated by K. Durr.

[26]

G. N. Milstein, Numerical Integration of Stochastic Differential Equations, vol. 313 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1995, Translated and revised from the 1988 Russian original. doi: 10.1007/978-94-015-8455-5.

[27]

G. N. Milstein and M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Scientific Computation, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-10063-9.

[28]

P. M. Morkisz and P. Przyby lowicz, Optimal pointwise approximation of SDE's from inexact information, J. Comput. Appl. Math., 324 (2017), 85-100, URL http://dx.doi.org/10.1016/j.cam.2017.04.023. doi: 10.1016/j.cam.2017.04.023.

[29]

P. Przyby lowicz, Minimal asymptotic error for one-point approximation of SDEs with timeirregular coefficients, J. Comput. Appl. Math., 282 (2015), 98-110, URL http://dx.doi.org/10.1016/j.cam.2015.01.003. doi: 10.1016/j.cam.2015.01.003.

[30]

P. Przyby lowicz, Optimal global approximation of SDEs with time-irregular coefficients in asymptotic setting, Appl. Math. Comput., 270 (2015), 441-457, URL http://dx.doi.org/10.1016/j.amc.2015.08.055. doi: 10.1016/j.amc.2015.08.055.

[31]

P. Przyby lowicz and P. Morkisz, Strong approximation of solutions of stochastic differential equations with time-irregular coefficients via randomized Euler algorithm, Appl. Numer. Math., 78 (2014), 80-94, URL http://dx.doi.org/10.1016/j.apnum.2013.12.003. doi: 10.1016/j.apnum.2013.12.003.

[32]

T. Rydén and M. Wiktorsson, On the simulation of iterated Itô integrals, Stochastic Process. Appl., 91 (2001), 151-168.  doi: 10.1016/S0304-4149(00)00053-3.

[33]

M. N. Spijker, Stability and Convergence of Finite-Difference Methods, vol. 1968 of Doctoral dissertation, University of Leiden, Rijksuniversiteit te Leiden, Leiden, 1968.

[34]

M. N. Spijker, On the structure of error estimates for finite-difference methods, Numer. Math., 18 (1971/72), 73-100.  doi: 10.1007/BF01398460.

[35]

G. Stengle, Numerical methods for systems with measurable coefficients, Appl. Math. Lett., 3 (1990), 25-29, URL http://dx.doi.org/10.1016/0893-9659(90)90040-I. doi: 10.1016/0893-9659(90)90040-I.

[36]

G. Stengle, Error analysis of a randomized numerical method, Numer. Math. , 70 (1995), 119-128, URL http://dx.doi.org/10.1007/s002110050113. doi: 10.1007/s002110050113.

[37]

F. Stummel, Approximation Methods in Analysis, Matematisk Institut, Aarhus Universitet, Aarhus, 1973, Lectures delivered during the spring term, 1973, Lecture Notes Series, No. 35.

[38]

J. F. Traub, G. W. Wasilkowski and H. Woźniakowski, Information-based Complexity, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1988, With contributions by A. G. Werschulz and T. Boult.

[39]

M. Wiktorsson, Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions, Ann. Appl. Probab., 11 (2001), 470-487.  doi: 10.1214/aoap/1015345301.

show all references

References:
[1]

W.-J. Beyn and R. Kruse, Two-sided error estimates for the stochastic theta method, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 389-407, URL http://dx.doi.org/10.3934/dcdsb.2010.14.389. doi: 10.3934/dcdsb.2010.14.389.

[2]

D. L. Burkholder, Martingale transforms, Ann. Math. Statist., 37 (1966), 1494-1504.  doi: 10.1214/aoms/1177699141.

[3]

J. M. C. Clark and R. J. Cameron, The maximum rate of convergence of discrete approximations for stochastic differential equations, in Stochastic Differential Systems (Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978), vol. 25 of Lecture Notes in Control and Information Sci., Springer, Berlin, 1980,162-171.

[4]

T. Daun, On the randomized solution of initial value problems, J. Complexity, 27 (2011), 300-311, URL http://dx.doi.org/10.1016/j.jco.2010.07.002. doi: 10.1016/j.jco.2010.07.002.

[5]

E. Emmrich, Discrete versions of Gronwall's lemma and their application to the numerical analysis of parabolic problems, TU Berlin, FB Mathematik, Preprint.

[6]

J. G. Gaines and T. J. Lyons, Random generation of stochastic area integrals, SIAM J. Appl. Math., 54 (1994), 1132-1146, URL http://dx.doi.org/10.1137/S0036139992235706. doi: 10.1137/S0036139992235706.

[7]

M. B. Giles and L. Szpruch, Antithetic multilevel Monte Carlo estimation for multidimensional SDEs without Lévy area simulation, Ann. Appl. Probab., 24 (2014), 1585-1620, URL http://dx.doi.org/10.1214/13-AAP957. doi: 10.1214/13-AAP957.

[8]

I. Gyöngy, A note on Euler's approximations, Potential Anal. , 8 (1998), 205-216, URL http://dx.doi.org/10.1023/A:1008605221617.

[9]

S. Haber, A modified Monte-Carlo quadrature, Math. Comp., 20 (1966), 361-368.  doi: 10.1090/S0025-5718-1966-0210285-0.

[10]

S. Haber, A modified Monte-Carlo quadrature. Ⅱ, Math. Comp., 21 (1967), 388-397.  doi: 10.1090/S0025-5718-1967-0234606-9.

[11]

S. Heinrich and B. Milla, The randomized complexity of initial value problems, J. Complexity, 24 (2008), 77-88.  doi: 10.1016/j.jco.2007.09.002.

[12]

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, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 1563-1576, URL http://dx.doi.org/10.1098/rspa.2010.0348. doi: 10.1098/rspa.2010.0348.

[13]

A. Jentzen and A. Neuenkirch, A random Euler scheme for Carathéodory differential equations, J. Comput. Appl. Math., 224 (2009), 346-359, URL http://dx.doi.org/10.1016/j.cam.2008.05.060. doi: 10.1016/j.cam.2008.05.060.

[14]

B. Kacewicz, Optimal solution of ordinary differential equations, J. Complexity, 3 (1987), 451-465, URL http://dx.doi.org/10.1016/0885-064X(87)90011-2. doi: 10.1016/0885-064X(87)90011-2.

[15]

B. Kacewicz, Asymptotic setting (revisited): Analysis of a boundary-value problem and a relation to a classical approximation result, J. Complexity, 20 (2004), 796-806, URL https://doi.org/10.1016/j.jco.2003.08.006. doi: 10.1016/j.jco.2003.08.006.

[16]

B. Kacewicz, Almost optimal solution of initial-value problems by randomized and quantum algorithms, J. Complexity, 22 (2006), 676-690, URL http://dx.doi.org/10.1016/j.jco.2006.03.001. doi: 10.1016/j.jco.2006.03.001.

[17]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, vol. 113 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, New York, 1991, URL https://doi.org/10.1007/978-1-4612-0949-2. doi: 10.1007/978-1-4612-0949-2.

[18]

P. E. Kloeden and A. Neuenkirch, The pathwise convergence of approximation schemes for stochastic differential equations, LMS J. Comput. Math., 10 (2007), 235-253, URL http://dx.doi.org/10.1112/S1461157000001388. doi: 10.1112/S1461157000001388.

[19]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, 3rd edition, Springer, Berlin, 1999. doi: 10.1007/978-3-662-12616-5.

[20]

R. Kruse, Characterization of bistability for stochastic multistep methods, BIT, 52 (2012), 109-140, URL http://dx.doi.org/10.1007/s10543-011-0341-5. doi: 10.1007/s10543-011-0341-5.

[21]

R. Kruse, Consistency and stability of a Milstein-Galerkin finite element scheme for semilinear SPDE, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014), 471-516, URL http://dx.doi.org/10.1007/s40072-014-0037-3. doi: 10.1007/s40072-014-0037-3.

[22]

R. Kruse and Y. Wu, Error analysis of randomized Runge-Kutta methods for differential equations with time-irregular coefficients, Comput. Methods Appl. Math., 17 (2017), 479-498.  doi: 10.1515/cmam-2016-0048.

[23]

X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood Publishing Limited, Chichester, 2008, URL http://dx.doi.org/10.1533/9780857099402. doi: 10.1533/9780857099402.

[24]

G. N. Milstein, Approximate integration of stochastic differential equations, Teor. Verojatnost. i Primenen., 19 (1974), 583-588, In Russian.

[25]

G. N. Milstein, Approximate integration of stochastic differential equations, Theory Probab. Appl., 19 (1975), 557-562, Translated by K. Durr.

[26]

G. N. Milstein, Numerical Integration of Stochastic Differential Equations, vol. 313 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1995, Translated and revised from the 1988 Russian original. doi: 10.1007/978-94-015-8455-5.

[27]

G. N. Milstein and M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Scientific Computation, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-10063-9.

[28]

P. M. Morkisz and P. Przyby lowicz, Optimal pointwise approximation of SDE's from inexact information, J. Comput. Appl. Math., 324 (2017), 85-100, URL http://dx.doi.org/10.1016/j.cam.2017.04.023. doi: 10.1016/j.cam.2017.04.023.

[29]

P. Przyby lowicz, Minimal asymptotic error for one-point approximation of SDEs with timeirregular coefficients, J. Comput. Appl. Math., 282 (2015), 98-110, URL http://dx.doi.org/10.1016/j.cam.2015.01.003. doi: 10.1016/j.cam.2015.01.003.

[30]

P. Przyby lowicz, Optimal global approximation of SDEs with time-irregular coefficients in asymptotic setting, Appl. Math. Comput., 270 (2015), 441-457, URL http://dx.doi.org/10.1016/j.amc.2015.08.055. doi: 10.1016/j.amc.2015.08.055.

[31]

P. Przyby lowicz and P. Morkisz, Strong approximation of solutions of stochastic differential equations with time-irregular coefficients via randomized Euler algorithm, Appl. Numer. Math., 78 (2014), 80-94, URL http://dx.doi.org/10.1016/j.apnum.2013.12.003. doi: 10.1016/j.apnum.2013.12.003.

[32]

T. Rydén and M. Wiktorsson, On the simulation of iterated Itô integrals, Stochastic Process. Appl., 91 (2001), 151-168.  doi: 10.1016/S0304-4149(00)00053-3.

[33]

M. N. Spijker, Stability and Convergence of Finite-Difference Methods, vol. 1968 of Doctoral dissertation, University of Leiden, Rijksuniversiteit te Leiden, Leiden, 1968.

[34]

M. N. Spijker, On the structure of error estimates for finite-difference methods, Numer. Math., 18 (1971/72), 73-100.  doi: 10.1007/BF01398460.

[35]

G. Stengle, Numerical methods for systems with measurable coefficients, Appl. Math. Lett., 3 (1990), 25-29, URL http://dx.doi.org/10.1016/0893-9659(90)90040-I. doi: 10.1016/0893-9659(90)90040-I.

[36]

G. Stengle, Error analysis of a randomized numerical method, Numer. Math. , 70 (1995), 119-128, URL http://dx.doi.org/10.1007/s002110050113. doi: 10.1007/s002110050113.

[37]

F. Stummel, Approximation Methods in Analysis, Matematisk Institut, Aarhus Universitet, Aarhus, 1973, Lectures delivered during the spring term, 1973, Lecture Notes Series, No. 35.

[38]

J. F. Traub, G. W. Wasilkowski and H. Woźniakowski, Information-based Complexity, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1988, With contributions by A. G. Werschulz and T. Boult.

[39]

M. Wiktorsson, Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions, Ann. Appl. Probab., 11 (2001), 470-487.  doi: 10.1214/aoap/1015345301.

Figure 1.  Numerical experiment for SDE (48): Step sizes versus $L^2$ errors
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Figure 2.  Numerical experiment for SDE (48): CPU time versus $L^2$ errors
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Table Listing 1.  A sample implementation of (9) in Python
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