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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.

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
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