January  2019, 18(1): 341-360. doi: 10.3934/cpaa.2019018

New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts

1. 

Center for Applied Mathematics, Tianjin University, Tianjin 300072, China

2. 

Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, United Kingdom

* Corresponding author

Received  January 2018 Revised  April 2018 Published  August 2018

In this paper, some new results on the the regularity of Kolmogorov equations associated to the infinite dimensional OU-process are obtained. As an application, the average $L^2$-error on $[0, T]$ of exponential integrator scheme for a range of semi-linear stochastic partial differential equations is derived, where the drift term is assumed to be Hölder continuous with respect to the Sobolev norm $\|·\|_{β}$ for some appropriate $β>0$. In addition, under a stronger condition on the drift, the strong convergence estimate is obtained, which covers the result of the SDEs with Hölder continuous drift.

Citation: Jianhai Bao, Xing Huang, Chenggui Yuan. New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts. Communications on Pure and Applied Analysis, 2019, 18 (1) : 341-360. doi: 10.3934/cpaa.2019018
References:
[1]

J. Bao, X. Huang and C. Yuan, Convergence Rate of Euler-Maruyama Scheme for SDEs with Rough Coefficients, arXiv: 1609.06080.

[2]

A. Barth and A. Lang, $L^p$ and almost sure convergence of a Milstein scheme for stochastic partial differential equations, Stochastic Process. Appl., 123 (2013), 1563-1587.  doi: 10.1016/j.spa.2013.01.003.

[3]

A. BarthA. Lang and Ch. Schwab, Multilevel Monte Carlo method for parabolic stochastic partial differential equations, BIT Numerical Mathematics, 53 (2013), 3-27.  doi: 10.1007/s10543-012-0401-5.

[4]

P. E. Chaudru de Raynal, Strong existence and uniqueness for stochastic differential equation with Hölder drift and degenerate noise, to appear in Ann. Inst. Henri Poincaré Probab. Stat.. doi: 10.1214/15-AIHP716.

[5]

A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp., 273 (2011), 89-117.  doi: 10.1090/S0025-5718-2010-02395-6.

[6]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[7]

G. Da PratoF. FlandoliE. Priola and M. Röckner, Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift, Ann. Probab., 41 (2013), 3306-3344.  doi: 10.1214/12-AOP763.

[8]

G. Da PratoF. FlandoliE. Priola and M. Röckner, Strong uniqueness for stochastic evolution equations with unbounded measurable drift term, J. Theor. Probab., 28 (2015), 1571-1600.  doi: 10.1007/s10959-014-0545-0.

[9]

G. Da Prato, F. Flandoli, M. Röckner and A. Yu, Veretennikov, Strong uniqueness for SDEs in Hilbert spaces with non-regular drift, arXiv: 1404.5418. doi: 10.1214/15-AOP1016.

[10]

E. Fedrizzi and F. Flandoli, Pathwise uniqueness and continuous dependence for SDEs with nonirregular drift, Stochastics, 83 (2011), 241-257.  doi: 10.1080/17442508.2011.553681.

[11]

T. E. Govindan, Mild solutions of neutral stochastic partial functional differential equations, International Journal of Stochastic Analysis, (2011), 186206. 

[12]

I. Gyöngy and N. Krylov, On the splitting-up method and stochastic partial differential equations, Ann. Probab., 31 (2003), 564-591.  doi: 10.1214/aop/1048516528.

[13]

I. Gyöngy and T. Martinez, On stochastic differential equations with locally unbounded drift, Czechoslovak Math.J., 51 (2001), 763-783.  doi: 10.1023/A:1013764929351.

[14]

I. Gyöngy and M. Rásonyi, A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients, Stochastic Process. Appl., 121 (2011), 2189-2200.  doi: 10.1016/j.spa.2011.06.008.

[15]

I. GyöngyS. Sabanis and D. Šiška, Convergence of tamed Euler schemes for a class of stochastic evolution equations, Stoch PDE: Anal. Comp., 4 (2016), 225-245.  doi: 10.1007/s40072-015-0057-7.

[16]

E. Hausenblas, Approximation for semilinear stochastic evolution equations, Potential Anal., 18 (2003), 141-186.  doi: 10.1023/A:1020552804087.

[17]

X. Huang and Z. Liao, The Euler-Maruyama method for (functional) SDEs with Hölder drift and $\alpha$-stable noise, Stoch. Anal. Appl., 36 (2018), 28-39.  doi: 10.1080/07362994.2017.1371037.

[18]

P. E. KloedenG. J. LordA. Neuenkirch and T. Shardlow, The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds, J. Comput. Appl. Math., 235 (2011), 1245-1260.  doi: 10.1016/j.cam.2010.08.011.

[19]

R. Kruse, Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise, IMA J. Numer. Anal., 34 (2014), 217-251.  doi: 10.1093/imanum/drs055.

[20]

N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Relat. Fields, 131 (2005), 154-196.  doi: 10.1007/s00440-004-0361-z.

[21]

A. LangP.-L. Chow and J. Potthoff, Almost sure convergence for a semidiscrete Milstein scheme for SPDEs of Zakai type, Stochastics, 82 (2010), 315-326.  doi: 10.1080/17442501003653497.

[22]

G. J. Lord and J. Rougemont, A numerical scheme for stochastic PDEs with gevrey regularity, IMA J. Numer. Anal., 24 (2004), 587-604.  doi: 10.1093/imanum/24.4.587.

[23]

G. J. Lord and T. Shardlow, Postprocessing for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 45 (2007), 870-899.  doi: 10.1137/050640138.

[24]

H.-L. Ngo and D. Taguchi, Strong rate of convergence for the Euler-Maruyama approximation of stochastic differential equations with irregular coefficients, Math. Comp., 85 (2016), 1793-1819.  doi: 10.1090/mcom3042.

[25]

H.-L. Ngo and D. Taguchi, On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients, arXiv: 1509.06532v1. doi: 10.1093/imanum/drw058.

[26]

O. M. Pamen and D. Taguchi, Strong rate of convergence for the Euler-Maruyama approximation of SDEs with Hölder continuous drift coefficient, arXiv: 1508.07513v1.

[27]

T. Shardlow, Numerical methods for stochastic parabolic PDEs, Numer. Funct. Anal. Optim., 20 (1999), 121-145.  doi: 10.1080/01630569908816884.

[28]

F.-Y. Wang, Gradient estimate and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift, J.Differential Equations, 260 (2016), 2792-2829.  doi: 10.1016/j.jde.2015.10.020.

[29]

F.-Y. Wang and X. Zhang, Degenerate SDEs in Hilbert spaces with rough drifts, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 18 (2015), 25 pp.  doi: 10.1142/S0219025715500265.

[30]

F.-Y. Wang and X. Zhang, Degenerate SDE with Hölder-Dini drift and non-Lipschitz noise coefficient, SIAM J. Math. Anal., 48 (2016), 2189-2226.  doi: 10.1137/15M1023671.

[31]

X. Zhang, Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients, Electron. J. Probab., 16 (2011), 1096-1116.  doi: 10.1214/EJP.v16-887.

[32]

X. Zhang, Strong solutions of SDEs with singural drift and Sobolev diffusion coefficients, Stoch. Proc. Appl., 115 (2005), 1805-1818.  doi: 10.1016/j.spa.2005.06.003.

[33]

A. K. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift, Mat. Sbornik, 93 (1974), 129-149. 

show all references

References:
[1]

J. Bao, X. Huang and C. Yuan, Convergence Rate of Euler-Maruyama Scheme for SDEs with Rough Coefficients, arXiv: 1609.06080.

[2]

A. Barth and A. Lang, $L^p$ and almost sure convergence of a Milstein scheme for stochastic partial differential equations, Stochastic Process. Appl., 123 (2013), 1563-1587.  doi: 10.1016/j.spa.2013.01.003.

[3]

A. BarthA. Lang and Ch. Schwab, Multilevel Monte Carlo method for parabolic stochastic partial differential equations, BIT Numerical Mathematics, 53 (2013), 3-27.  doi: 10.1007/s10543-012-0401-5.

[4]

P. E. Chaudru de Raynal, Strong existence and uniqueness for stochastic differential equation with Hölder drift and degenerate noise, to appear in Ann. Inst. Henri Poincaré Probab. Stat.. doi: 10.1214/15-AIHP716.

[5]

A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp., 273 (2011), 89-117.  doi: 10.1090/S0025-5718-2010-02395-6.

[6]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[7]

G. Da PratoF. FlandoliE. Priola and M. Röckner, Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift, Ann. Probab., 41 (2013), 3306-3344.  doi: 10.1214/12-AOP763.

[8]

G. Da PratoF. FlandoliE. Priola and M. Röckner, Strong uniqueness for stochastic evolution equations with unbounded measurable drift term, J. Theor. Probab., 28 (2015), 1571-1600.  doi: 10.1007/s10959-014-0545-0.

[9]

G. Da Prato, F. Flandoli, M. Röckner and A. Yu, Veretennikov, Strong uniqueness for SDEs in Hilbert spaces with non-regular drift, arXiv: 1404.5418. doi: 10.1214/15-AOP1016.

[10]

E. Fedrizzi and F. Flandoli, Pathwise uniqueness and continuous dependence for SDEs with nonirregular drift, Stochastics, 83 (2011), 241-257.  doi: 10.1080/17442508.2011.553681.

[11]

T. E. Govindan, Mild solutions of neutral stochastic partial functional differential equations, International Journal of Stochastic Analysis, (2011), 186206. 

[12]

I. Gyöngy and N. Krylov, On the splitting-up method and stochastic partial differential equations, Ann. Probab., 31 (2003), 564-591.  doi: 10.1214/aop/1048516528.

[13]

I. Gyöngy and T. Martinez, On stochastic differential equations with locally unbounded drift, Czechoslovak Math.J., 51 (2001), 763-783.  doi: 10.1023/A:1013764929351.

[14]

I. Gyöngy and M. Rásonyi, A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients, Stochastic Process. Appl., 121 (2011), 2189-2200.  doi: 10.1016/j.spa.2011.06.008.

[15]

I. GyöngyS. Sabanis and D. Šiška, Convergence of tamed Euler schemes for a class of stochastic evolution equations, Stoch PDE: Anal. Comp., 4 (2016), 225-245.  doi: 10.1007/s40072-015-0057-7.

[16]

E. Hausenblas, Approximation for semilinear stochastic evolution equations, Potential Anal., 18 (2003), 141-186.  doi: 10.1023/A:1020552804087.

[17]

X. Huang and Z. Liao, The Euler-Maruyama method for (functional) SDEs with Hölder drift and $\alpha$-stable noise, Stoch. Anal. Appl., 36 (2018), 28-39.  doi: 10.1080/07362994.2017.1371037.

[18]

P. E. KloedenG. J. LordA. Neuenkirch and T. Shardlow, The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds, J. Comput. Appl. Math., 235 (2011), 1245-1260.  doi: 10.1016/j.cam.2010.08.011.

[19]

R. Kruse, Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise, IMA J. Numer. Anal., 34 (2014), 217-251.  doi: 10.1093/imanum/drs055.

[20]

N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Relat. Fields, 131 (2005), 154-196.  doi: 10.1007/s00440-004-0361-z.

[21]

A. LangP.-L. Chow and J. Potthoff, Almost sure convergence for a semidiscrete Milstein scheme for SPDEs of Zakai type, Stochastics, 82 (2010), 315-326.  doi: 10.1080/17442501003653497.

[22]

G. J. Lord and J. Rougemont, A numerical scheme for stochastic PDEs with gevrey regularity, IMA J. Numer. Anal., 24 (2004), 587-604.  doi: 10.1093/imanum/24.4.587.

[23]

G. J. Lord and T. Shardlow, Postprocessing for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 45 (2007), 870-899.  doi: 10.1137/050640138.

[24]

H.-L. Ngo and D. Taguchi, Strong rate of convergence for the Euler-Maruyama approximation of stochastic differential equations with irregular coefficients, Math. Comp., 85 (2016), 1793-1819.  doi: 10.1090/mcom3042.

[25]

H.-L. Ngo and D. Taguchi, On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients, arXiv: 1509.06532v1. doi: 10.1093/imanum/drw058.

[26]

O. M. Pamen and D. Taguchi, Strong rate of convergence for the Euler-Maruyama approximation of SDEs with Hölder continuous drift coefficient, arXiv: 1508.07513v1.

[27]

T. Shardlow, Numerical methods for stochastic parabolic PDEs, Numer. Funct. Anal. Optim., 20 (1999), 121-145.  doi: 10.1080/01630569908816884.

[28]

F.-Y. Wang, Gradient estimate and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift, J.Differential Equations, 260 (2016), 2792-2829.  doi: 10.1016/j.jde.2015.10.020.

[29]

F.-Y. Wang and X. Zhang, Degenerate SDEs in Hilbert spaces with rough drifts, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 18 (2015), 25 pp.  doi: 10.1142/S0219025715500265.

[30]

F.-Y. Wang and X. Zhang, Degenerate SDE with Hölder-Dini drift and non-Lipschitz noise coefficient, SIAM J. Math. Anal., 48 (2016), 2189-2226.  doi: 10.1137/15M1023671.

[31]

X. Zhang, Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients, Electron. J. Probab., 16 (2011), 1096-1116.  doi: 10.1214/EJP.v16-887.

[32]

X. Zhang, Strong solutions of SDEs with singural drift and Sobolev diffusion coefficients, Stoch. Proc. Appl., 115 (2005), 1805-1818.  doi: 10.1016/j.spa.2005.06.003.

[33]

A. K. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift, Mat. Sbornik, 93 (1974), 129-149. 

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