Stochastic Taylor expansions for the expectation of functionals of diffusion processes.

*(English)*Zbl 1065.60068The author considers systems of stochastic ordinary differential equations (SODEs) both in Itô and Stratonovich formulation, driven by a multi-dimensional Wiener process. In order to develop and analyse weak approximation methods, one needs stochastic Taylor expansions of \({\mathbf E}f(X_t)\), the expectation of a functional \(f\) of the solution \(X_t\) to the SODEs in both formulations. In this well-written article a representation of \({\mathbf E}f(X_t)\) in terms of truncated Taylor expansions and remainder terms is proved. Further, estimates of the remainder terms are given. The representations and the proofs are based on multi-coloured stochastic rooted tree theory. This generalizes the rooted tree theory well-known in the deterministic analysis of Runge-Kutta methods. In particular, the Taylor expansions derived here coincide with those derived in the deterministic setting if the diffusion term vanishes.

Reviewer: Evelyn Buckwar (Berlin)

##### MSC:

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

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

60J60 | Diffusion processes |

41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |

##### Keywords:

stochastic ordinary differential equations; stochastic Taylor expansion; diffusion processes; stochastic multi-coloured rooted tree theory; Itô calculus; Stratonovich calculus; weak approximations
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\textit{A. Rößler}, Stochastic Anal. Appl. 22, No. 6, 1553--1576 (2004; Zbl 1065.60068)

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