##
**On the splitting-up method and stochastic partial differential equations.**
*(English)*
Zbl 1028.60058

The authors consider stochastic partial differential equations essentially (generalisations are also given) of the form
\[
du(t,x) = Lu(t,x)dt + M_ku(t,x) \circ dw_t^k, \quad x \in \mathbb{R}^d, \;t\in [0,T], \tag{1}
\]
and some initial data \(u_0 = u_0(x)\). Here \(w_t^k\) are \(k\) independent one-dimensional Wiener processes and equation (1) is to be interpreted in the Stratonovich sense. The \(M_k\)’s are first-order operators and \(L\) is a second-order elliptic operator, all acting on functions defined on \(R^d\). The authors propose and analyse splitting-up methods for the numerical approximation of the solution of (1). By stretching out the time-scale in a certain way they can reformulate the splitting-up method as a stochastic equation and apply stochastic calculus for their proofs (previous proofs relied on semigroup theory). One of their theorems says that for each \(T>0\) and \(p>0\) there is a constant \(N\) such that \(E \max_{t \in T_n} \|u_n(t) -u(t)\|_0^p \leq N/h^p\) for all \(n>1\), where \(u_n(t)\) is the numerical approximation to (1), \(E\) the expectation, \(T_n\) a grid on \([0,T]\) and \(\|.\|_0\) the usual \(L_2\)-norm on \(R^d\). Auxiliary results, the proof of the above theorem, as well as generalisations of it are presented in Sections 2 to 5. The last section provides an application to nonlinear filtering.

Reviewer: Evelyn Buckwar (Berlin)

### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |

65C30 | Numerical solutions to stochastic differential and integral equations |

35R60 | PDEs with randomness, stochastic partial differential equations |

93E25 | Computational methods in stochastic control (MSC2010) |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

93E11 | Filtering in stochastic control theory |

### Keywords:

nonlinear filtering; Zakai equation; splitting-up method; numerical approximation; strong convergence; stochastic initial value problem; error bounds
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\textit{I. Gyöngy} and \textit{N. Krylov}, Ann. Probab. 31, No. 2, 564--591 (2003; Zbl 1028.60058)

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### References:

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