The projection method for coarse-graining stochastic differential equations does not generally match stochastic averaging under time-scale separation, and this paper gives conditions and counterexamples showing when they agree and when they do not.
Mimicking an it\^ o process by a solution of a stochastic differential equation
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Coarse graining of stochastic differential equations: averaging and projection method
The projection method for coarse-graining stochastic differential equations does not generally match stochastic averaging under time-scale separation, and this paper gives conditions and counterexamples showing when they agree and when they do not.