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September  2019, 11(3): 325-359. doi: 10.3934/jgm.2019018

## A symmetry-adapted numerical scheme for SDEs

 1 Institute for Applied Mathematics and HCM, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany 2 Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy

* Corresponding author: stefania.ugolini@unimmi.it

Received  May 2017 Revised  July 2019 Published  August 2019

Fund Project: The first author is supported by the German Research Foundation (DFG) via CRC 1060.

We propose a geometric numerical analysis of SDEs admitting Lie symmetries which allows us to individuate a symmetry adapted coordinates system where the given SDE puts in evidence notable invariant properties. An approximation scheme preserving the symmetry properties of the equation is introduced. Our algorithmic procedure is applied to the family of general linear SDEs for which two theoretical estimates of the numerical forward error are established.

Citation: Francesco C. De Vecchi, Andrea Romano, Stefania Ugolini. A symmetry-adapted numerical scheme for SDEs. Journal of Geometric Mechanics, 2019, 11 (3) : 325-359. doi: 10.3934/jgm.2019018
##### References:
 [1] S. Albeverio, F. C. De Vecchi, P. Morando and S. Ugolini, Weak symmetries of stochastic differential equations driven by semimartingales with jumps, arXiv preprint, arXiv: 1904.10963. Google Scholar [2] S. Albeverio, F. C. De Vecchi and S. Ugolini, Entropy chaos and bose-einstein condensation, J. Stat. Phys., 168 (2017), 483-507.  doi: 10.1007/s10955-017-1820-0.  Google Scholar [3] C. Anton, J. Deng and Y. S. Wong, Weak symplectic schemes for stochastic Hamiltonian equations, Electron. Trans. Numer. Anal., 43 (2014/15), 1-20.   Google Scholar [4] N. Bou-Rabee and H. Owhadi, Stochastic variational integrators, IMA J. Numer. Anal., 29 (2009), 421-443.  doi: 10.1093/imanum/drn018.  Google Scholar [5] N. Bou-Rabee and H. Owhadi, Long-run accuracy of variational integrators in the stochastic context, SIAM J. Numer. Anal., 48 (2010), 278-297.  doi: 10.1137/090758842.  Google Scholar [6] R. Campoamor-Stursberg, M. A. Rodríguez and P. Winternitz, Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations, J. Phys. A, 49 (2016), 035201, 21pp. doi: 10.1088/1751-8113/49/3/035201.  Google Scholar [7] C. Chen, D. Cohen and J. Hong, Conservative methods for stochastic differential equations with a conserved quantity, Int. J. Numer. Anal. Model., 13 (2016), 435-456.   Google Scholar [8] F. C. De Vecchi, Lie Symmetry Analysis and Geometrical Methods for Finite and Infinite Dimensional Stochastic Differential Equations, PhD thesis, 2017. Google Scholar [9] F. C. De Vecchi, P. Morando and S. Ugolini, A note on symmetries of diffusions within a martingale problem approach, Stochastics and Dynamics, 19 (2019), 1950011, 21 pp. doi: 10.1142/S0219493719500114.  Google Scholar [10] F. C. De Vecchi, P. Morando and S. Ugolini, Reduction and reconstruction of stochastic differential equations via symmetries, J. Math. Phys., 57 (2016), 123508, 22pp. doi: 10.1063/1.4973197.  Google Scholar [11] F. C. De Vecchi, P. Morando and S. Ugolini, Symmetries of stochastic differential equations: A geometric approach, J. Math. Phys, 57 (2016), 063504, 17pp. doi: 10.1063/1.4953374.  Google Scholar [12] V. Dorodnitsyn, Applications of Lie Groups to Difference Equations, vol. 8 of Differential and Integral Equations and Their Applications, CRC Press, Boca Raton, FL, 2011.   Google Scholar [13] K. Ebrahimi-Fard, A. Lundervold, S. J. A. Malham, H. Munthe-Kaas and A. Wiese, Algebraic structure of stochastic expansions and efficient simulation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468 (2012), 2361-2382.  doi: 10.1098/rspa.2012.0024.  Google Scholar [14] N. B.-L. Eckhard Platen, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-13694-8.  Google Scholar [15] A. Friedman, Stochastic Differential Equations and Applications. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975, Probability and Mathematical Statistics, Vol. 28.  Google Scholar [16] P. Friz and S. Riedel, Convergence rates for the full Gaussian rough paths, Ann. Inst. Henri Poincaré Probab. Stat., 50 (2014), 154-194.  doi: 10.1214/12-AIHP507.  Google Scholar [17] P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014, With an introduction to regularity structures. doi: 10.1007/978-3-319-08332-2.  Google Scholar [18] G. Gaeta and N. R. Quintero, Lie-point symmetries and stochastic differential equations, J. Phys. A, 32 (1999), 8485-8505.  doi: 10.1088/0305-4470/32/48/310.  Google Scholar [19] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, vol. 31 of Springer Series in Computational Mathematics, 2nd edition, Springer-Verlag, Berlin, 2006, Structure-preserving algorithms for ordinary differential equations.  Google Scholar [20] D. B. Hernández and R. Spigler, $A$-stability of Runge-Kutta methods for systems with additive noise, BIT, 32 (1992), 620-633.  doi: 10.1007/BF01994846.  Google Scholar [21] D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38 (2000), 753–769 (electronic). doi: 10.1137/S003614299834736X.  Google Scholar [22] D. J. Higham and P. E. Kloeden, Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math., 101 (2005), 101-119.  doi: 10.1007/s00211-005-0611-8.  Google Scholar [23] D. J. Higham, X. Mao and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal., 45 (2007), 592–609 (electronic). doi: 10.1137/060658138.  Google Scholar [24] D. D. Holm and T. M. Tyranowski, Variational principles for stochastic soliton dynamics, Proc. A., 472 (2016), 20150827, 24pp. doi: 10.1098/rspa.2015.0827.  Google Scholar [25] J. Hong, S. Zhai and J. Zhang, Discrete gradient approach to stochastic differential equations with a conserved quantity, SIAM J. Numer. Anal., 49 (2011), 2017-2038.  doi: 10.1137/090771880.  Google Scholar [26] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009.  Google Scholar [27] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications of Mathematics (New York), Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar [28] R. Kozlov, Symmetries of systems of stochastic differential equations with diffusion matrices of full rank, J. Phys. A, 43 (2010), 245201, 16pp. doi: 10.1088/1751-8113/43/24/245201.  Google Scholar [29] J.-A. Lázaro-Camí and J.-P. Ortega, Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations, Stoch. Dyn., 9 (2009), 1-46.  doi: 10.1142/S0219493709002531.  Google Scholar [30] B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, vol. 14 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2004.  Google Scholar [31] P. Lescot and J.-C. Zambrini, Probabilistic deformation of contact geometry, diffusion processes and their quadratures, in Seminar on Stochastic Analysis, Random Fields and Applications V, vol. 59 of Progr. Probab., Birkhäuser, Basel, 2008,203–226. doi: 10.1007/978-3-7643-8458-6_12.  Google Scholar [32] D. Levi, P. Olver, Z. Thomova and P. Winternitz (eds.), Symmetries and Integrability of Difference Equations, vol. 381 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2011, https://doi.org/10.1007/978-3-319-56666-5, Lectures from the Summer School (Séminaire de Máthematiques Supérieures) held at the Université de Montréal, Montréal, QC, June 8–21, 2008. doi: 10.1017/CBO9780511997136.  Google Scholar [33] D. Levi and P. Winternitz, Continuous symmetries of difference equations, J. Phys. A, 39 (2006), R1–R63. doi: 10.1088/0305-4470/39/2/R01.  Google Scholar [34] S. J. A. Malham and A. Wiese, Stochastic Lie group integrators, SIAM J. Sci. Comput., 30 (2008), 597-617.  doi: 10.1137/060666743.  Google Scholar [35] S. V. Meleshko, Y. N. Grigoriev, N. K. Ibragimov and V. F. Kovalev, Symmetries of Integro-Differential Equations: With Applications in Mechanics and Plasma Physics, vol. 806, Springer Science & Business Media, 2010. doi: 10.1007/978-90-481-3797-8.  Google Scholar [36] G. N. Milstein, E. Platen and H. Schurz, Balanced implicit methods for stiff stochastic systems, SIAM J. Numer. Anal., 35 (1998), 1010–1019 (electronic). doi: 10.1137/S0036142994273525.  Google Scholar [37] G. N. Milstein, Y. M. Repin and M. V. Tretyakov, Numerical methods for stochastic systems preserving symplectic structure, SIAM J. Numer. Anal., 40 (2002), 1583–1604 (electronic). doi: 10.1137/S0036142901395588.  Google Scholar [38] L. M. Morato and S. Ugolini, Stochastic description of a bose-einstein condensate, Ann. H. Poincaré, 12 (2011), 1601-1612.  doi: 10.1007/s00023-011-0116-1.  Google Scholar [39] B. Øksendal, Stochastic Differential Equations, Sixth edition, Universitext, Springer-Verlag, Berlin, 2003, An introduction with applications. doi: 10.1007/978-3-642-14394-6.  Google Scholar [40] P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, New York, 1993.  Google Scholar [41] É. Pardoux and P. E. Protter, A two-sided stochastic integral and its calculus, Probab. Theory Related Fields, 76 (1987), 15-49.  doi: 10.1007/BF00390274.  Google Scholar [42] G. R. W. Quispel and R. I. McLachlan, Special issue on geometric numerical integration of differential equations, Journal of Physics A: Mathematical and General, 39 (2006), front matter (3 pp.). doi: 10.1088/0305-4470/39/19/E01.  Google Scholar [43] L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales, Vol. 2, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000, Itô calculus, Reprint of the second (1994) edition. doi: 10.1017/CBO9781107590120.  Google Scholar [44] Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267.  doi: 10.1137/S0036142992228409.  Google Scholar [45] M. Tao, H. Owhadi and J. E. Marsden, Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging, Multiscale Model. Simul., 8 (2010), 1269-1324.  doi: 10.1137/090771648.  Google Scholar [46] A. Tocino, Mean-square stability of second-order Runge-Kutta methods for stochastic differential equations, J. Comput. Appl. Math., 175 (2005), 355-367.  doi: 10.1016/j.cam.2004.05.019.  Google Scholar [47] L. Wang, J. Hong, R. Scherer and F. Bai, Dynamics and variational integrators of stochastic Hamiltonian systems, Int. J. Numer. Anal. Model., 6 (2009), 586-602.   Google Scholar [48] C. Yuan and X. Mao, Stability in distribution of numerical solutions for stochastic differential equations, Stochastic Anal. Appl., 22 (2004), 1133-1150.  doi: 10.1081/SAP-200026423.  Google Scholar

show all references

##### References:
 [1] S. Albeverio, F. C. De Vecchi, P. Morando and S. Ugolini, Weak symmetries of stochastic differential equations driven by semimartingales with jumps, arXiv preprint, arXiv: 1904.10963. Google Scholar [2] S. Albeverio, F. C. De Vecchi and S. Ugolini, Entropy chaos and bose-einstein condensation, J. Stat. Phys., 168 (2017), 483-507.  doi: 10.1007/s10955-017-1820-0.  Google Scholar [3] C. Anton, J. Deng and Y. S. Wong, Weak symplectic schemes for stochastic Hamiltonian equations, Electron. Trans. Numer. Anal., 43 (2014/15), 1-20.   Google Scholar [4] N. Bou-Rabee and H. Owhadi, Stochastic variational integrators, IMA J. Numer. Anal., 29 (2009), 421-443.  doi: 10.1093/imanum/drn018.  Google Scholar [5] N. Bou-Rabee and H. Owhadi, Long-run accuracy of variational integrators in the stochastic context, SIAM J. Numer. Anal., 48 (2010), 278-297.  doi: 10.1137/090758842.  Google Scholar [6] R. Campoamor-Stursberg, M. A. Rodríguez and P. Winternitz, Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations, J. Phys. A, 49 (2016), 035201, 21pp. doi: 10.1088/1751-8113/49/3/035201.  Google Scholar [7] C. Chen, D. Cohen and J. Hong, Conservative methods for stochastic differential equations with a conserved quantity, Int. J. Numer. Anal. Model., 13 (2016), 435-456.   Google Scholar [8] F. C. De Vecchi, Lie Symmetry Analysis and Geometrical Methods for Finite and Infinite Dimensional Stochastic Differential Equations, PhD thesis, 2017. Google Scholar [9] F. C. De Vecchi, P. Morando and S. Ugolini, A note on symmetries of diffusions within a martingale problem approach, Stochastics and Dynamics, 19 (2019), 1950011, 21 pp. doi: 10.1142/S0219493719500114.  Google Scholar [10] F. C. De Vecchi, P. Morando and S. Ugolini, Reduction and reconstruction of stochastic differential equations via symmetries, J. Math. Phys., 57 (2016), 123508, 22pp. doi: 10.1063/1.4973197.  Google Scholar [11] F. C. De Vecchi, P. Morando and S. Ugolini, Symmetries of stochastic differential equations: A geometric approach, J. Math. Phys, 57 (2016), 063504, 17pp. doi: 10.1063/1.4953374.  Google Scholar [12] V. Dorodnitsyn, Applications of Lie Groups to Difference Equations, vol. 8 of Differential and Integral Equations and Their Applications, CRC Press, Boca Raton, FL, 2011.   Google Scholar [13] K. Ebrahimi-Fard, A. Lundervold, S. J. A. Malham, H. Munthe-Kaas and A. Wiese, Algebraic structure of stochastic expansions and efficient simulation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468 (2012), 2361-2382.  doi: 10.1098/rspa.2012.0024.  Google Scholar [14] N. B.-L. Eckhard Platen, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-13694-8.  Google Scholar [15] A. Friedman, Stochastic Differential Equations and Applications. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975, Probability and Mathematical Statistics, Vol. 28.  Google Scholar [16] P. Friz and S. Riedel, Convergence rates for the full Gaussian rough paths, Ann. Inst. Henri Poincaré Probab. Stat., 50 (2014), 154-194.  doi: 10.1214/12-AIHP507.  Google Scholar [17] P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014, With an introduction to regularity structures. doi: 10.1007/978-3-319-08332-2.  Google Scholar [18] G. Gaeta and N. R. Quintero, Lie-point symmetries and stochastic differential equations, J. Phys. A, 32 (1999), 8485-8505.  doi: 10.1088/0305-4470/32/48/310.  Google Scholar [19] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, vol. 31 of Springer Series in Computational Mathematics, 2nd edition, Springer-Verlag, Berlin, 2006, Structure-preserving algorithms for ordinary differential equations.  Google Scholar [20] D. B. Hernández and R. Spigler, $A$-stability of Runge-Kutta methods for systems with additive noise, BIT, 32 (1992), 620-633.  doi: 10.1007/BF01994846.  Google Scholar [21] D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38 (2000), 753–769 (electronic). doi: 10.1137/S003614299834736X.  Google Scholar [22] D. J. Higham and P. E. Kloeden, Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math., 101 (2005), 101-119.  doi: 10.1007/s00211-005-0611-8.  Google Scholar [23] D. J. Higham, X. Mao and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal., 45 (2007), 592–609 (electronic). doi: 10.1137/060658138.  Google Scholar [24] D. D. Holm and T. M. Tyranowski, Variational principles for stochastic soliton dynamics, Proc. A., 472 (2016), 20150827, 24pp. doi: 10.1098/rspa.2015.0827.  Google Scholar [25] J. Hong, S. Zhai and J. Zhang, Discrete gradient approach to stochastic differential equations with a conserved quantity, SIAM J. Numer. Anal., 49 (2011), 2017-2038.  doi: 10.1137/090771880.  Google Scholar [26] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009.  Google Scholar [27] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications of Mathematics (New York), Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar [28] R. Kozlov, Symmetries of systems of stochastic differential equations with diffusion matrices of full rank, J. Phys. A, 43 (2010), 245201, 16pp. doi: 10.1088/1751-8113/43/24/245201.  Google Scholar [29] J.-A. Lázaro-Camí and J.-P. Ortega, Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations, Stoch. Dyn., 9 (2009), 1-46.  doi: 10.1142/S0219493709002531.  Google Scholar [30] B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, vol. 14 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2004.  Google Scholar [31] P. Lescot and J.-C. Zambrini, Probabilistic deformation of contact geometry, diffusion processes and their quadratures, in Seminar on Stochastic Analysis, Random Fields and Applications V, vol. 59 of Progr. Probab., Birkhäuser, Basel, 2008,203–226. doi: 10.1007/978-3-7643-8458-6_12.  Google Scholar [32] D. Levi, P. Olver, Z. Thomova and P. Winternitz (eds.), Symmetries and Integrability of Difference Equations, vol. 381 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2011, https://doi.org/10.1007/978-3-319-56666-5, Lectures from the Summer School (Séminaire de Máthematiques Supérieures) held at the Université de Montréal, Montréal, QC, June 8–21, 2008. doi: 10.1017/CBO9780511997136.  Google Scholar [33] D. Levi and P. Winternitz, Continuous symmetries of difference equations, J. Phys. A, 39 (2006), R1–R63. doi: 10.1088/0305-4470/39/2/R01.  Google Scholar [34] S. J. A. Malham and A. Wiese, Stochastic Lie group integrators, SIAM J. Sci. Comput., 30 (2008), 597-617.  doi: 10.1137/060666743.  Google Scholar [35] S. V. Meleshko, Y. N. Grigoriev, N. K. Ibragimov and V. F. Kovalev, Symmetries of Integro-Differential Equations: With Applications in Mechanics and Plasma Physics, vol. 806, Springer Science & Business Media, 2010. doi: 10.1007/978-90-481-3797-8.  Google Scholar [36] G. N. Milstein, E. Platen and H. Schurz, Balanced implicit methods for stiff stochastic systems, SIAM J. Numer. Anal., 35 (1998), 1010–1019 (electronic). doi: 10.1137/S0036142994273525.  Google Scholar [37] G. N. Milstein, Y. M. Repin and M. V. Tretyakov, Numerical methods for stochastic systems preserving symplectic structure, SIAM J. Numer. Anal., 40 (2002), 1583–1604 (electronic). doi: 10.1137/S0036142901395588.  Google Scholar [38] L. M. Morato and S. Ugolini, Stochastic description of a bose-einstein condensate, Ann. H. Poincaré, 12 (2011), 1601-1612.  doi: 10.1007/s00023-011-0116-1.  Google Scholar [39] B. Øksendal, Stochastic Differential Equations, Sixth edition, Universitext, Springer-Verlag, Berlin, 2003, An introduction with applications. doi: 10.1007/978-3-642-14394-6.  Google Scholar [40] P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, New York, 1993.  Google Scholar [41] É. Pardoux and P. E. Protter, A two-sided stochastic integral and its calculus, Probab. Theory Related Fields, 76 (1987), 15-49.  doi: 10.1007/BF00390274.  Google Scholar [42] G. R. W. Quispel and R. I. McLachlan, Special issue on geometric numerical integration of differential equations, Journal of Physics A: Mathematical and General, 39 (2006), front matter (3 pp.). doi: 10.1088/0305-4470/39/19/E01.  Google Scholar [43] L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales, Vol. 2, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000, Itô calculus, Reprint of the second (1994) edition. doi: 10.1017/CBO9781107590120.  Google Scholar [44] Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267.  doi: 10.1137/S0036142992228409.  Google Scholar [45] M. Tao, H. Owhadi and J. E. Marsden, Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging, Multiscale Model. Simul., 8 (2010), 1269-1324.  doi: 10.1137/090771648.  Google Scholar [46] A. Tocino, Mean-square stability of second-order Runge-Kutta methods for stochastic differential equations, J. Comput. Appl. Math., 175 (2005), 355-367.  doi: 10.1016/j.cam.2004.05.019.  Google Scholar [47] L. Wang, J. Hong, R. Scherer and F. Bai, Dynamics and variational integrators of stochastic Hamiltonian systems, Int. J. Numer. Anal. Model., 6 (2009), 586-602.   Google Scholar [48] C. Yuan and X. Mao, Stability in distribution of numerical solutions for stochastic differential equations, Stochastic Anal. Appl., 22 (2004), 1133-1150.  doi: 10.1081/SAP-200026423.  Google Scholar
Strong and weak errors with $t \in [0.1, 1]$ and stepsize $h = 0.025$
Strong and weak errors with $t \in [0.1, 1]$ and stepsize $h = 0.01$
Strong and weak errors with $t = 0.5$ and step number $N = [10, 80]$
Total variation distance with $t = 0.5$ and $h \in [10, 80]$
$X_t$ strong and weak errors with $t \in [0.1, 1]$ and stepsize $h = 0.025$
$Y_t$ strong and weak errors with $t \in [0.1, 1]$ and stepsize $h = 0.025$
$X_t$ strong and weak errors with $T = 1$ and step number $N = [10, 100]$
$Y_t$ strong and weak errors with $T = 1$ and step number $N = [10, 100]$
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