Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Taylor expansions of solutions of stochastic partial differential equations

Pages: 515 - 557, Volume 14, Issue 2, September 2010      doi:10.3934/dcdsb.2010.14.515

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Arnulf Jentzen - Faculty of Mathematics, Bielefeld University, Universitätsstr. 25, 33501 Bielefeld, Germany (email)

Abstract: 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.

Keywords:  Taylor expansions, Stochastic partial differential equations.
Mathematics Subject Classification:  Primary: 60H15; Secondary: 35K90.

Received: August 2009;      Revised: January 2010;      Available Online: June 2010.