We consider a one-dimensional stochastic differential equation driven by a Wiener process, where the diffusion coefficient depends on an ergodic fast process. The averaging principle is satisfied: it is well-known that the slow component converges in distribution to the solution of an averaged equation, with generator determined by averaging the square of the diffusion coefficient.
We propose a new version of the averaging principle, where the solution is interpreted as the sum of two terms: one depending on the average of the diffusion coefficient, the other giving fluctuations around that average. Both the average and fluctuation terms contribute to the limit, which illustrates why it is required to average the square of the diffusion coefficient to find the limit behavior.
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@article{CRMATH_2022__360_G3_265_0,
author = {Charles-Edouard Br\'ehier},
title = {The averaging principle for stochastic differential equations driven by a {Wiener} process revisited},
journal = {Comptes Rendus. Math\'ematique},
pages = {265--273},
publisher = {Acad\'emie des sciences, Paris},
volume = {360},
year = {2022},
doi = {10.5802/crmath.297},
language = {en},
}
TY - JOUR AU - Charles-Edouard Bréhier TI - The averaging principle for stochastic differential equations driven by a Wiener process revisited JO - Comptes Rendus. Mathématique PY - 2022 SP - 265 EP - 273 VL - 360 PB - Académie des sciences, Paris DO - 10.5802/crmath.297 LA - en ID - CRMATH_2022__360_G3_265_0 ER -
Charles-Edouard Bréhier. The averaging principle for stochastic differential equations driven by a Wiener process revisited. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 265-273. doi : 10.5802/crmath.297. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.297/
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