Abstract
Based on the shuffle product expansion of exponential Lie series in terms of a Philip Hall basis for the stochastic differential equations of jump-diffusion type, we can establish Stratonovich–Taylor–Hall (STH) schemes. However, the STHr scheme converges only at order r in the mean-square sense. In order to have the almost sure Stratonovich–Taylor–Hall (ASTH) schemes, we have to include all the terms related to multiple Poissonian integrals as the moments of multiple Poissonian integrals always have lower orders of magnitudes as compared with those of multiple Brownian integrals.
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Li, C.W., Liu, X.Q. Almost Sure Convergence of the Numerical Discretization of Stochastic Jump Diffusions. Acta Applicandae Mathematicae 62, 225–244 (2000). https://doi.org/10.1023/A:1006495115904
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DOI: https://doi.org/10.1023/A:1006495115904