# zbMATH — the first resource for mathematics

Stochastic Taylor expansions for the expectation of functionals of diffusion processes. (English) Zbl 1065.60068
The 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.

##### 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)
Full Text:
##### References:
 [1] Arnold L., Stochastic Differential Equations (1974) · Zbl 0278.60039 [2] Burrage K., SIAM J. Numer. Anal. 38 (5) pp 1626– (2000) · Zbl 0983.65006 [3] Burrage K., Appl. Numer. Math. 22 (1) pp 81– (1996) · Zbl 0868.65101 [4] Butcher J. C., The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods (1987) · Zbl 0616.65072 [5] Dynkin E.B., Markov Processes (1965) [6] Ikeda N., Kodansha Ltd.: Tokyo, in: Stochastic Differential Equations and Diffusion Processes (1989) [7] Karatzas I., Brownian Motion and Stochastic Calculus (1999) · Zbl 0734.60060 [8] Kloeden P.E., Applications of Mathematics 23, in: Numerical Solution of Stochastic Differential Equations (1999) [9] Øksendal B., Applications of Mathematics 21, in: Stochastic Differential Equations (1998) [10] Platen E., Probab. Math. Statist. 3 pp 37– (1982) [11] Stroock D.W., Multidimensional Diffusion Processes (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.