September  2010, 14(2): 515-557. doi: 10.3934/dcdsb.2010.14.515

Taylor expansions of solutions of stochastic partial differential equations

1. 

Faculty of Mathematics, Bielefeld University, Universitätsstr. 25, 33501 Bielefeld, Germany

Received  August 2009 Revised  January 2010 Published  June 2010

The solution of a stochastic partial differential equation (SPDE) of evolutionary type is with respect to a reasonable state space in general not a semimartingale anymore and does therefore in general not satisfy an Itô formula like the solution of a finite dimensional stochastic ordinary differential equation. Consequently, stochastic Taylor expansions of the solution of a SPDE can not be derived by an iterated application of Itô's formula. Recently, in [Jentzen and Kloeden, Ann. Probab. 38 (2010), no. 2, 532-569] in the case of SPDEs with additive noise an alternative approach for deriving Taylor expansions has been introduced by using the mild formulation of the SPDE and by an appropriate recursion technique. This method is used in this article to derive Taylor expansions of arbitrarily high order of the solution of a SPDE with non-additive noise.
Citation: Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[1]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[2]

Quan Zhou, Yabing Sun. High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021233

[3]

Rainer Buckdahn, Ingo Bulla, Jin Ma. Pathwise Taylor expansions for Itô random fields. Mathematical Control and Related Fields, 2011, 1 (4) : 437-468. doi: 10.3934/mcrf.2011.1.437

[4]

Yaozhong Hu, Yanghui Liu, David Nualart. Taylor schemes for rough differential equations and fractional diffusions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3115-3162. doi: 10.3934/dcdsb.2016090

[5]

Min Yang, Guanggan Chen. Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1565-1581. doi: 10.3934/dcdsb.2019240

[6]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021020

[7]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295

[8]

Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006

[9]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[10]

Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203

[11]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks and Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014

[12]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221

[13]

Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061

[14]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control and Related Fields, 2021, 11 (4) : 797-828. doi: 10.3934/mcrf.2020047

[15]

Juan C. Cortés, Sandra E. Delgadillo-Alemán, Roberto A. Kú-Carrillo, Rafael J. Villanueva. Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022079

[16]

Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118

[17]

Ishak Alia. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control and Related Fields, 2020, 10 (4) : 785-826. doi: 10.3934/mcrf.2020020

[18]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[19]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473

[20]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (314)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]