TAYLOR EXPANSIONS OF SOLUTIONS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH ADDITIVE NOISE

By Arnulf Jentzen, Peter E. Kloeden. 2010

67 citations recorded by the source corpus.

The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinite-dimensional Brownian motion is in general not a semi-martingale anymore and does in general not satisfy an Ito formula like the solution of a finite-dimensional stochastic ordinary differential equation (SODE). In particular, it is not possible to derive stochastic Taylor expansions as for the solution of a SODE using an iterated application of the Ito formula. Consequently, until recently, only low order numerical approximation results for such a SPDE have been available. Here, the fact that the solution of a SPDE driven by additive noise can be interpreted in the mild sense with integrals involving the exponential of the dominant linear operator in the SPDE provides an alternative approach for deriving stochastic Taylor expansions for the solution of such a SPDE. Essentially, the exponential factor has a mollifying effect and ensures that all integrals take values in the Hilbert space under consideration. The iteration of such integrals allows us to derive stochastic Taylor expansions of arbitrarily high order, which are robust in the sense that they also hold for other types of driving noise processes such as fractional Brownian motion. Combinatorial concepts of trees and woods provide a compact formulation of the Taylor expansions.

Subfields: Finance, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/23aab725e79a43f42e336c5bffea8f1867f04ab6

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In Finance

1. Arnulf Jentzen
2. Peter E. Kloeden
3. Tony Shardlow
4. M. Roeckner
5. Giuseppe Da Prato
6. Xuerong Mao
7. Sotirios Sabanis
8. Martin Hairer

In Statistical and Nonlinear Physics

1. Tony Shardlow
2. Peter E. Kloeden
3. Arnulf Jentzen
4. Giuseppe Da Prato

Citations in the corpus

• The Milstein Scheme for Stochastic Delay Differential Equations Without Using Anticipative Calculus (2012, methodology): “Following the approach used by Jentzen & Kloeden for random ordinary differential equations [7, 6] and stochastic partial differential equations [8], we use deterministic Taylor expansions of the coefficient functions, with lower order expansions being inserted into the right hand side of higher ord”
• A mild Itô formula for SPDEs (2010, background): “Similar and related stochastic Taylor expansions for SPDEs can be found in Buckdahn & Ma [11], Bayer & Teichmann [4], Conus [15], Jentzen & Kloeden [54], Buckdahn, Bulla & Ma [10] and Jentzen [49].”
• The Numerical Approximation of Stochastic Partial Differential Equations (2009, methodology): “Numerical approximations based on the new Taylor expansions for SPDEs, which are reviewed in Section 5 in this article, are considered in [61, 62, 63, 64, 67, 68].”