A Drift-Filtered Approach to Diffusion Estimation for Multiscale Processes

Semiparametric Drift and Diffusion Estimation for Multiscale Diffusions

“…where the integrals are being approximated by a quadrature rule and t is chosen appropriately (in fact, we have approximated the Lebesgue integral via the trapezoidal rule). If one, however, uses a rectangular-method with the left corner node instead, this estimator coincides with the estimator proposed in [16] for estimating the effective diffusion coefficient based on observations of the slow component of a fast/slow system. We emphasize, that a crucial assumption on the estimator in the aforementioned work is that the effective drift is known a priori.…”

“…Figure 3(a) seems to suggest that the optimal subsampling rate is given by the local extremum of the estimator as function of the subsampling. However, this behavior is not true in general, see for instance the numerical examples in [47,16] that reveal different behaviors. Recall, that the optimal subsampling rate is in general unknown.…”

“…We conclude this section with a remark on a recently proposed estimator for constant diffusion coefficients [16] that can also be derived using the approach we introduce here. To this end, assume that the coarse-grained equation takes the form:…”

“…This assumption is very restrictive and makes the estimator unfeasible for most practical applications. A further limitation of the estimator in [16] is that it applies only in situations where the noise in the coarse-grained equation is additive (constant diffusion coefficient). Conversely, the methodology proposed here aims to estimate multiple parameters in both the drift and the diffusion coefficients.…”

“…with an appropriate constant c n ∈ R that depends not only on n but also on u and t, and which vanishes in the limit as n → ∞. Here Q n denotes the quadrature operator of the trapezoidal rule as defined in (16). The actual error of the trapezoidal rule depends on the regularity of the integrand.…”