Product expansion for stochastic jump diffusions and its application to numerical approximation
\textit{H. J. Sussmann} [Sel. Pap. 7th Int. Symp. Math. Theory Networks Syst., Stockholm 1985, 323-335 (1986; Zbl 0619.93021)] expressed the Chen series as the infinite product of the exponential Lie series in terms of a Philip Hall basis for the Lie algebra generated by the ordinary differential operator. Based on this result, one can obtain numerical schemes of an appropriate order by truncating up to a certain number of terms of the exponential Lie series. The aim of this paper is to develop a new product expansion of the exponential Lie series for jump diffusions with a single Poisson noise while that with multi-Poisson noises is expressed as shuffle products which implicitly represent the ordinary part of products of multiple stochastic integrals. On the basis of this expansion, the authors can construct the Stratonovich-Taylor-Hall (STH) schemes which are proved to be more efficient than Stratonovich-Taylor schemes, as they involve only the minimum number of multiple stochastic integrals, which can be regarded as systems of stochastic jump-diffusion equations and approximated by a lower-order scheme with an appropriate stepsize to ensure the necessary accuracy. Mean-square convergence of the STH schemes is shown, and numerical results are provided to illustrate the proposed approach.
scientific article; zbMATH DE number 1367885
scientific article