August 2018, 23(6): 2217-2243. doi: 10.3934/dcdsb.2018232

On the mild Itô formula in Banach spaces

1. 

Korteweg-de Vries Instituut, University of Amsterdam, Amsterdam, the Netherlands

2. 

Seminar for Applied Mathematics, Department of Mathematics, ETH Zürich, Zürich, Switzerland

* Corresponding author: Sonja Cox

Received  December 2016 Revised  July 2017 Published  July 2018

Fund Project: This work is partly financed by the NWO-research programme VENI Vernieuwingsimpuls with project number 639:031:549. It is also partly financed by SNSF-Research project 200021_156603 "Numerical approximations of nonlinear stochastic ordinary and partial differential equations"

The mild Itô formula proposed in Theorem 1 in [Da Prato, G., Jentzen, A., & Röckner, M., A mild Ito formula for SPDEs, arXiv: 1009.3526 (2012), To appear in the Trans. Amer. Math. Soc.] has turned out to be a useful instrument to study solutions and numerical approximations of stochastic partial differential equations (SPDEs) which are formulated as stochastic evolution equations (SEEs) on Hilbert spaces. In this article we generalize this mild Itô formula so that it is applicable to stopping times instead of deterministic time points and so that it is applicable to solutions and numerical approximations of SPDEs which are formulated as SEEs on UMD (unconditional martingale differences) Banach spaces. These generalizations are especially useful for proving essentially sharp weak convergence rates for numerical approximations of SPDEs such as stochastic heat equations with nonlinear diffusion coefficients.

Citation: Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232
References:
[1]

A. Andersson, A. Jentzen and R. Kurniawan, Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values, arXiv: 1512.06899, (2016), 35 pages. Revision requested from J. Math. Anal. Appl.

[2]

B. Birnir, The Kolmogorov-Obukhov statistical theory of turbulence, J. Nonlinear Sci., 23 (2013), 657-688. doi: 10.1007/s00332-012-9164-z.

[3]

B. Birnir, The Kolmogorov-Obukhov Theory of Turbulence, SpringerBriefs in Mathematics. Springer, New York, 2013. A mathematical theory of turbulence. doi: 10.1007/978-1-4614-6262-0.

[4]

Z. Brzeźniak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629. doi: 10.1090/S0002-9947-06-03923-7.

[5]

Z. BrzeźniakJ. M. A. M. v. NeervenM. C. Veraar and L. Weis, Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation, J. Differential Equations, 245 (2008), 30-58. doi: 10.1016/j.jde.2008.03.026.

[6]

E. CelledoniD. Cohen and B. Owren, Symmetric exponential integrators with an application to the cubic Schrödinger equation, Found. Comput. Math., 8 (2008), 303-317. doi: 10.1007/s10208-007-9016-7.

[7]

D. Cohen and L. Gauckler, One-stage exponential integrators for nonlinear Schrödinger equations over long times, BIT, 52 (2012), 877-903. doi: 10.1007/s10543-012-0385-1.

[8]

G. Da Prato, A. Jentzen and M. Röckner, A mild Ito formula for SPDEs, arXiv: 1009.3526, (2012), 39 pages. To appear in Trans. Amer. Math. Soc.

[9]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[10]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.

[11]

D. FilipovićS. Tappe and J. Teichmann, Term structure models driven by Wiener processes and Poisson measures: existence and positivity, SIAM J. Financial Math., 1 (2010), 523-554. doi: 10.1137/090758593.

[12]

P. Harms, D. Stefanovits, J. Teichmann and M. Wüthrich, Consistent recalibration of yield curve models, arXiv: 1502.02926v1, (2015), 40 pages. To appear in Math. Finance.

[13]

M. Hefter, A. Jentzen and R. Kurniawan, Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces, arXiv: 1612.03209, (2016), 51 pages.

[14]

A. Jentzen and R. Kurniawan, Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients, arXiv: 1501.03539, (2015), 51 pages.

[15]

A. Jentzen and P. Pušnik, Exponential moments for numerical approximations of stochastic partial differential equations, arXiv: 1609.07031, (2016), 44 pages. Revision requested from Stoch. Partial Differ. Equ. Anal. Comput.

[16]

G. Kallianpur and J. Xiong, Stochastic models of environmental pollution, Adv. in Appl. Probab., 26 (1994), 377-403. doi: 10.2307/1427442.

[17]

M. A. Kouritzin and H. Long, Convergence of Markov chain approximations to stochastic reaction-diffusion equations, Ann. Appl. Probab., 12 (2002), 1039-1070. doi: 10.1214/aoap/1031863180.

[18]

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.

[19]

G. J. Lord and A. Tambue, Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise, IMA J. Numer. Anal., 33 (2013), 515-543. doi: 10.1093/imanum/drr059.

[20]

J.-C. Mourrat and H. Weber, Convergence of the two-dimensional dynamic Ising-Kac model to $Φ^4_2$, Comm. Pure Appl. Math., 70 (2017), 717-812. doi: 10.1002/cpa.21655.

[21]

J. v. Neerven, γ-radonifying operators—a survey, In The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, vol. 44 of Proc. Centre Math. Appl. Austral. Nat. Univ. Austral. Nat. Univ., Canberra, 2010, 1-61.

[22]

J. v. NeervenM. Veraar and L. Weis, Stochastic evolution equations in UMD Banach spaces, J. Funct. Anal., 255 (2008), 940-993. doi: 10.1016/j.jfa.2008.03.015.

[23]

J. v. NeervenM. Veraar and L. Weis, Stochastic integration in Banach spaces - a survey, Progress in Probability, 68 (2015), 297-332. doi: 10.1007/978-3-0348-0909-2_11.

[24]

K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3. Academic Press, Inc., New York-London, 1967.

[25]

M. Sauer and W. Stannat, Lattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition, Math. Comp., 84 (2015), 743-766. doi: 10.1090/S0025-5718-2014-02873-1.

[26]

X. Wang, An exponential integrator scheme for time discretization of nonlinear stochastic wave equation, J. Sci. Comput., 64 (2015), 234-263. doi: 10.1007/s10915-014-9931-0.

show all references

References:
[1]

A. Andersson, A. Jentzen and R. Kurniawan, Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values, arXiv: 1512.06899, (2016), 35 pages. Revision requested from J. Math. Anal. Appl.

[2]

B. Birnir, The Kolmogorov-Obukhov statistical theory of turbulence, J. Nonlinear Sci., 23 (2013), 657-688. doi: 10.1007/s00332-012-9164-z.

[3]

B. Birnir, The Kolmogorov-Obukhov Theory of Turbulence, SpringerBriefs in Mathematics. Springer, New York, 2013. A mathematical theory of turbulence. doi: 10.1007/978-1-4614-6262-0.

[4]

Z. Brzeźniak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629. doi: 10.1090/S0002-9947-06-03923-7.

[5]

Z. BrzeźniakJ. M. A. M. v. NeervenM. C. Veraar and L. Weis, Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation, J. Differential Equations, 245 (2008), 30-58. doi: 10.1016/j.jde.2008.03.026.

[6]

E. CelledoniD. Cohen and B. Owren, Symmetric exponential integrators with an application to the cubic Schrödinger equation, Found. Comput. Math., 8 (2008), 303-317. doi: 10.1007/s10208-007-9016-7.

[7]

D. Cohen and L. Gauckler, One-stage exponential integrators for nonlinear Schrödinger equations over long times, BIT, 52 (2012), 877-903. doi: 10.1007/s10543-012-0385-1.

[8]

G. Da Prato, A. Jentzen and M. Röckner, A mild Ito formula for SPDEs, arXiv: 1009.3526, (2012), 39 pages. To appear in Trans. Amer. Math. Soc.

[9]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[10]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.

[11]

D. FilipovićS. Tappe and J. Teichmann, Term structure models driven by Wiener processes and Poisson measures: existence and positivity, SIAM J. Financial Math., 1 (2010), 523-554. doi: 10.1137/090758593.

[12]

P. Harms, D. Stefanovits, J. Teichmann and M. Wüthrich, Consistent recalibration of yield curve models, arXiv: 1502.02926v1, (2015), 40 pages. To appear in Math. Finance.

[13]

M. Hefter, A. Jentzen and R. Kurniawan, Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces, arXiv: 1612.03209, (2016), 51 pages.

[14]

A. Jentzen and R. Kurniawan, Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients, arXiv: 1501.03539, (2015), 51 pages.

[15]

A. Jentzen and P. Pušnik, Exponential moments for numerical approximations of stochastic partial differential equations, arXiv: 1609.07031, (2016), 44 pages. Revision requested from Stoch. Partial Differ. Equ. Anal. Comput.

[16]

G. Kallianpur and J. Xiong, Stochastic models of environmental pollution, Adv. in Appl. Probab., 26 (1994), 377-403. doi: 10.2307/1427442.

[17]

M. A. Kouritzin and H. Long, Convergence of Markov chain approximations to stochastic reaction-diffusion equations, Ann. Appl. Probab., 12 (2002), 1039-1070. doi: 10.1214/aoap/1031863180.

[18]

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.

[19]

G. J. Lord and A. Tambue, Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise, IMA J. Numer. Anal., 33 (2013), 515-543. doi: 10.1093/imanum/drr059.

[20]

J.-C. Mourrat and H. Weber, Convergence of the two-dimensional dynamic Ising-Kac model to $Φ^4_2$, Comm. Pure Appl. Math., 70 (2017), 717-812. doi: 10.1002/cpa.21655.

[21]

J. v. Neerven, γ-radonifying operators—a survey, In The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, vol. 44 of Proc. Centre Math. Appl. Austral. Nat. Univ. Austral. Nat. Univ., Canberra, 2010, 1-61.

[22]

J. v. NeervenM. Veraar and L. Weis, Stochastic evolution equations in UMD Banach spaces, J. Funct. Anal., 255 (2008), 940-993. doi: 10.1016/j.jfa.2008.03.015.

[23]

J. v. NeervenM. Veraar and L. Weis, Stochastic integration in Banach spaces - a survey, Progress in Probability, 68 (2015), 297-332. doi: 10.1007/978-3-0348-0909-2_11.

[24]

K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3. Academic Press, Inc., New York-London, 1967.

[25]

M. Sauer and W. Stannat, Lattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition, Math. Comp., 84 (2015), 743-766. doi: 10.1090/S0025-5718-2014-02873-1.

[26]

X. Wang, An exponential integrator scheme for time discretization of nonlinear stochastic wave equation, J. Sci. Comput., 64 (2015), 234-263. doi: 10.1007/s10915-014-9931-0.

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